Piecewise Continuous Function, Integral of an Even or Odd Function
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Real Fourier Series
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Complex Fourier Series
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Parseval Identity
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- Real Parseval Identity - Introduction
- Exercise 1
- Generalised Real Parseval Identity
- Exercise 2
- Exercise 3
- Complex Parseval Identity - Introduction
- Generalised Complex Parseval Identity
- Exercise 4 - Part a
- Exercise 4 - Part b
- Exercise 4 - Part c
- Exercise 5
- Exercise 6 - Part a
- Exercise 6 - Part b
- Exercise 6 - Part c

Riemann Lebesgue Lemma
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Dirichlet's Theorem
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Even and Odd Extensions
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Differentiation and Integration of Fourier Series
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- Piecewise Continuously Differentable Function
- Term by Term Differentiation of Fourier Series
- Term by Term Integration of Fourier Series
- Exercise 1
- Exercise 2 - Part a
- Exercise 2 - Part b
- Exercise 2 - Part c
- Exercise 2 - Part d
- Exercise 2 - Part e
- Exercise 2 - Part f
- Exercise 3 - Part a
- Exercise 3 - Part b
- Exercise 3 - Part c
- Exercise 4
- Exercise 5 - Part a
- Exercise 5 - Part b
- Exercise 6
- Exercise 7 - Part a
- Exercise 7 - Part b
- Exercise 8 - Part a
- Exercise 8 - Part b
- Exercise 9 - Part a
- Exercise 9 - Part b
- Exercise 10 - Part a
- Exercise 10 - Part b
- Decay Rate of Fourier Coefficients Theorem - Part 1
- Decay Rate of Fourier Coefficients Theorem - Part 2
- Exercise 11
- Exercise 12

Uniform Convergence of Fourier Series
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Fourier Series on a General Interval
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- Fourier Series on a General Interval + Example
- Exercise 1
- Exercise 2 - Part a
- Exercise 2 - Part b
- Exercise 3 - Part a
- Exercise 3 - Part b
- Exercise 3 - Part c
- Exercise 4
- Exercise 5
- Exercise 6 - Part a
- Exercise 6 - Part b
- Exercise 6- Part c
- Exercise 6 - Part d
- Sine and Cosine Series of a Function
- Exercise 7
- Exercise 8

Summarizing Exercises
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[{"Name":"Piecewise Continuous Function, Integral of an Even or Odd Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Piecewise Continuous Function","Duration":"3m 5s","ChapterTopicVideoID":27517,"CourseChapterTopicPlaylistID":294449,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.685","Text":"Before we get into Fourier series,"},{"Start":"00:02.685 ","End":"00:06.240","Text":"we better review what is a piecewise continuous function."},{"Start":"00:06.240 ","End":"00:08.295","Text":"It\u0027s important for Fourier series,"},{"Start":"00:08.295 ","End":"00:12.930","Text":"and start straightaway with an example of function here\u0027s its graph,"},{"Start":"00:12.930 ","End":"00:15.330","Text":"which is defined on the interval 0,"},{"Start":"00:15.330 ","End":"00:20.265","Text":"2 real valued and it\u0027s defined in 3 pieces as follows."},{"Start":"00:20.265 ","End":"00:21.975","Text":"Here it\u0027s x^2,"},{"Start":"00:21.975 ","End":"00:24.120","Text":"here it\u0027s 1 plus x,"},{"Start":"00:24.120 ","End":"00:26.955","Text":"and here is 2 plus cosine x,"},{"Start":"00:26.955 ","End":"00:28.665","Text":"gets in 3 pieces."},{"Start":"00:28.665 ","End":"00:32.025","Text":"Now formal definition of piecewise continuous,"},{"Start":"00:32.025 ","End":"00:35.775","Text":"on an interval, finite interval a, b,"},{"Start":"00:35.775 ","End":"00:39.170","Text":"it could also be an open interval or a half-open interval,"},{"Start":"00:39.170 ","End":"00:42.195","Text":"if there is a finite sequence of points,"},{"Start":"00:42.195 ","End":"00:47.480","Text":"starting at a and ending at b, first of all,"},{"Start":"00:47.480 ","End":"00:53.560","Text":"f is continuous on each of the subintervals x_k to x_k plus 1."},{"Start":"00:53.560 ","End":"00:57.920","Text":"It\u0027s continuous here from 0-1 open interval."},{"Start":"00:57.920 ","End":"00:59.930","Text":"It\u0027s continuous on the interval 1,"},{"Start":"00:59.930 ","End":"01:02.795","Text":"2, it\u0027s continuous on the interval 2, 3."},{"Start":"01:02.795 ","End":"01:08.944","Text":"Secondly, it also has limits at the inner points, the one-sided limits."},{"Start":"01:08.944 ","End":"01:14.090","Text":"Like at 1, it has a left limit and it has a right limit even though they\u0027re not equal."},{"Start":"01:14.090 ","End":"01:15.500","Text":"Similarly at 2,"},{"Start":"01:15.500 ","End":"01:18.965","Text":"it has a left limit and it has a right limit."},{"Start":"01:18.965 ","End":"01:24.080","Text":"Similarly at the endpoints and that\u0027s the third condition that it should"},{"Start":"01:24.080 ","End":"01:30.080","Text":"have one-sided limits on the right here and on the left here."},{"Start":"01:30.080 ","End":"01:32.300","Text":"Unless it\u0027s the open interval,"},{"Start":"01:32.300 ","End":"01:35.750","Text":"then you\u0027d forget that condition or part of it if it\u0027s half-open."},{"Start":"01:35.750 ","End":"01:39.095","Text":"There\u0027s also a definition for an infinite interval, we\u0027ll come to that."},{"Start":"01:39.095 ","End":"01:43.155","Text":"But let\u0027s straightaway do an example exercise."},{"Start":"01:43.155 ","End":"01:46.220","Text":"In each of the following we have to check if f is"},{"Start":"01:46.220 ","End":"01:49.595","Text":"piecewise continuous, there\u0027s 2 examples."},{"Start":"01:49.595 ","End":"01:52.300","Text":"In the first one, if we look at it,"},{"Start":"01:52.300 ","End":"01:55.505","Text":"it\u0027s fairly clear that it is piecewise continuous."},{"Start":"01:55.505 ","End":"02:00.265","Text":"It\u0027s continuous on the open interval from 0-1 and from 1-2,"},{"Start":"02:00.265 ","End":"02:02.550","Text":"and the points we mentioned are 0,"},{"Start":"02:02.550 ","End":"02:04.120","Text":"1, and 2."},{"Start":"02:04.120 ","End":"02:07.130","Text":"It\u0027s also got a left limit at 1,"},{"Start":"02:07.130 ","End":"02:10.055","Text":"right limit at 1, and so on."},{"Start":"02:10.055 ","End":"02:12.114","Text":"The answer is yes."},{"Start":"02:12.114 ","End":"02:16.790","Text":"But the answer to b where it\u0027s 1 over x and 0,"},{"Start":"02:16.790 ","End":"02:19.610","Text":"the answer here is no,"},{"Start":"02:19.610 ","End":"02:22.850","Text":"because one of the conditions isn\u0027t met."},{"Start":"02:22.850 ","End":"02:27.320","Text":"The limit on the right at 0 is the limit of 1"},{"Start":"02:27.320 ","End":"02:32.380","Text":"over x as x goes to 0 from the right not to infinity and we want it to be finite,"},{"Start":"02:32.380 ","End":"02:34.665","Text":"so the answer here is no."},{"Start":"02:34.665 ","End":"02:38.345","Text":"Like I said, there\u0027s a definition for infinite interval,"},{"Start":"02:38.345 ","End":"02:42.320","Text":"and what we do is we say that it\u0027s continuous on"},{"Start":"02:42.320 ","End":"02:48.395","Text":"an infinite interval if on each finite subinterval, it\u0027s piecewise continuous."},{"Start":"02:48.395 ","End":"02:53.780","Text":"For example, the floor function which is a step function is piecewise"},{"Start":"02:53.780 ","End":"02:59.210","Text":"continuous on any subinterval that\u0027s finite and therefore the whole real line."},{"Start":"02:59.210 ","End":"03:04.160","Text":"This is the floor function and it\u0027s written with a funny bracket like this."},{"Start":"03:04.160 ","End":"03:06.870","Text":"That\u0027s enough for this clip."}],"ID":28711},{"Watched":false,"Name":"Integral of an Even or Odd Function","Duration":"5m 10s","ChapterTopicVideoID":27516,"CourseChapterTopicPlaylistID":294449,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.365","Text":"In this clip, we\u0027ll talk about the definite integrals of even and odd functions."},{"Start":"00:04.365 ","End":"00:06.810","Text":"They will be useful later on."},{"Start":"00:06.810 ","End":"00:10.710","Text":"Some definitions, and you know this is just a reminder."},{"Start":"00:10.710 ","End":"00:12.540","Text":"A function f is called even,"},{"Start":"00:12.540 ","End":"00:16.905","Text":"if f(x) is f(minus x) for all x and it\u0027s called"},{"Start":"00:16.905 ","End":"00:22.230","Text":"odd if f(x) is minus f( minus x) for all x."},{"Start":"00:22.230 ","End":"00:24.816","Text":"Now some properties,"},{"Start":"00:24.816 ","End":"00:25.875","Text":"first of all,"},{"Start":"00:25.875 ","End":"00:30.600","Text":"regarding multiplication and division of odd and even functions."},{"Start":"00:30.600 ","End":"00:33.855","Text":"If f is even and g is even,"},{"Start":"00:33.855 ","End":"00:37.875","Text":"then the product and the quotient are even."},{"Start":"00:37.875 ","End":"00:41.955","Text":"If f is even and g is odd or vice versa,"},{"Start":"00:41.955 ","End":"00:46.545","Text":"then f times g and f over g are both odd."},{"Start":"00:46.545 ","End":"00:50.078","Text":"Lastly, if we have an odd and"},{"Start":"00:50.078 ","End":"00:53.835","Text":"an odd function then the product and the quotient are even."},{"Start":"00:53.835 ","End":"00:55.740","Text":"There\u0027s 2 ways of thinking about it,"},{"Start":"00:55.740 ","End":"01:03.030","Text":"1 is to think of addition where even plus even is even and even plus odd is odd,"},{"Start":"01:03.030 ","End":"01:06.810","Text":"or you can think of even like plus 1 and odd"},{"Start":"01:06.810 ","End":"01:11.235","Text":"like minus 1 and then you have multiplication or division."},{"Start":"01:11.235 ","End":"01:15.208","Text":"Positive times negative is negative,"},{"Start":"01:15.208 ","End":"01:19.770","Text":"and negative times negative is positive and so on, so whatever."},{"Start":"01:19.770 ","End":"01:22.470","Text":"Now, addition and subtraction."},{"Start":"01:22.470 ","End":"01:27.270","Text":"If you add or subtract 2 even functions the result is even,"},{"Start":"01:27.270 ","End":"01:31.110","Text":"and if you add or subtract 2 odd functions the result is odd."},{"Start":"01:31.110 ","End":"01:34.590","Text":"But there is no rule for adding an even to an odd,"},{"Start":"01:34.590 ","End":"01:37.365","Text":"or an odd to an even."},{"Start":"01:37.365 ","End":"01:40.545","Text":"Lastly, absolute value."},{"Start":"01:40.545 ","End":"01:42.390","Text":"If f is odd,"},{"Start":"01:42.390 ","End":"01:44.760","Text":"then absolute value of f is even."},{"Start":"01:44.760 ","End":"01:48.570","Text":"If f is even then the absolute value of f is even because"},{"Start":"01:48.570 ","End":"01:52.230","Text":"it\u0027s not true for a general function that the absolute value is even,"},{"Start":"01:52.230 ","End":"01:54.480","Text":"it\u0027s true for odd and for even functions."},{"Start":"01:54.480 ","End":"01:56.010","Text":"Now, let\u0027s practice these,"},{"Start":"01:56.010 ","End":"01:57.495","Text":"we\u0027ll have an exercise."},{"Start":"01:57.495 ","End":"02:03.405","Text":"Determine if each of the following functions is even or odd or neither."},{"Start":"02:03.405 ","End":"02:07.170","Text":"F(x) equals sin^2 x times cosine x."},{"Start":"02:07.170 ","End":"02:12.000","Text":"What we have here is because sine is odd and cosine is even we have odd squared,"},{"Start":"02:12.000 ","End":"02:16.035","Text":"which is odd times odd, which is even and even times even is even."},{"Start":"02:16.035 ","End":"02:21.190","Text":"In part b, we have the absolute value of sine x over 1 plus cosine^22 x."},{"Start":"02:21.190 ","End":"02:25.640","Text":"Its absolute value of odd over 1 plus even squared."},{"Start":"02:25.640 ","End":"02:27.890","Text":"Absolute value of odd is even,"},{"Start":"02:27.890 ","End":"02:29.525","Text":"1 is even,"},{"Start":"02:29.525 ","End":"02:31.123","Text":"and even times even is even,"},{"Start":"02:31.123 ","End":"02:32.510","Text":"and even plus even is even,"},{"Start":"02:32.510 ","End":"02:35.785","Text":"and even over even is even, so that\u0027s the answer."},{"Start":"02:35.785 ","End":"02:41.805","Text":"Now, x^3 plus x^2 arctanx over cosine hyperbolic of x."},{"Start":"02:41.805 ","End":"02:45.285","Text":"In case you\u0027ve forgotten what cosine hyperbolic of x is,"},{"Start":"02:45.285 ","End":"02:48.510","Text":"this is the definition and it\u0027s clear that"},{"Start":"02:48.510 ","End":"02:51.840","Text":"it\u0027s an even function because if you replace x by minus x,"},{"Start":"02:51.840 ","End":"02:54.750","Text":"we just switched the order here, nothing changes."},{"Start":"02:54.750 ","End":"02:58.710","Text":"What we have is x^3 is odd,"},{"Start":"02:58.710 ","End":"03:01.080","Text":"x^2 is even,"},{"Start":"03:01.080 ","End":"03:03.585","Text":"arc tangent is odd,"},{"Start":"03:03.585 ","End":"03:07.990","Text":"because tangent is odd and cosine hyperbolic here is even,"},{"Start":"03:07.990 ","End":"03:12.380","Text":"so we have odd even times odd is odd over even and odd plus odd is odd,"},{"Start":"03:12.380 ","End":"03:14.685","Text":"odd over even is odd."},{"Start":"03:14.685 ","End":"03:19.815","Text":"Now, the integral of an even or odd function on a symmetric interval."},{"Start":"03:19.815 ","End":"03:23.505","Text":"Symmetric interval is like the interval from minus L to L,"},{"Start":"03:23.505 ","End":"03:26.130","Text":"from minus something to the same something."},{"Start":"03:26.130 ","End":"03:34.200","Text":"The integral of an odd function from minus L to L is 0 and if f is even,"},{"Start":"03:34.200 ","End":"03:37.410","Text":"then the integral from minus L to L is"},{"Start":"03:37.410 ","End":"03:42.090","Text":"twice the integral from 0 to L. This can be useful."},{"Start":"03:42.090 ","End":"03:45.075","Text":"Let\u0027s just do an exercise involving these."},{"Start":"03:45.075 ","End":"03:47.685","Text":"We have 3 integrals to compute."},{"Start":"03:47.685 ","End":"03:50.010","Text":"In part A, we have"},{"Start":"03:50.010 ","End":"03:56.820","Text":"x^2021 times cosine of x. X to the power of an odd number is an odd function."},{"Start":"03:56.820 ","End":"03:58.530","Text":"Cosine is an even,"},{"Start":"03:58.530 ","End":"04:01.230","Text":"odd times even is odd,"},{"Start":"04:01.230 ","End":"04:03.990","Text":"so the integral is 0."},{"Start":"04:03.990 ","End":"04:07.365","Text":"The integral of an odd function on a symmetric interval is 0."},{"Start":"04:07.365 ","End":"04:12.150","Text":"Part B, absolute value of x is even,"},{"Start":"04:12.150 ","End":"04:15.420","Text":"so e to the power of this will be even."},{"Start":"04:15.420 ","End":"04:19.515","Text":"Any function of an even function is an even function."},{"Start":"04:19.515 ","End":"04:22.650","Text":"Sine x is an odd function."},{"Start":"04:22.650 ","End":"04:25.560","Text":"Even times odd is odd,"},{"Start":"04:25.560 ","End":"04:28.035","Text":"so the integral is 0 again."},{"Start":"04:28.035 ","End":"04:30.990","Text":"Part c, the integral is an even function."},{"Start":"04:30.990 ","End":"04:35.725","Text":"Absolute value of x is even x^2 is even so each of the x^2 is even."},{"Start":"04:35.725 ","End":"04:41.955","Text":"We can use the other formula and take it as twice the integral from 0 to Pi."},{"Start":"04:41.955 ","End":"04:43.740","Text":"Now, between 0 and Pi,"},{"Start":"04:43.740 ","End":"04:46.620","Text":"absolute value of x is the same as x."},{"Start":"04:46.620 ","End":"04:48.840","Text":"We have 2x,"},{"Start":"04:48.840 ","End":"04:51.525","Text":"put the 2 inside, 2x into the x^2,"},{"Start":"04:51.525 ","End":"04:53.870","Text":"because 2x is a derivative of x^2,"},{"Start":"04:53.870 ","End":"04:58.175","Text":"this whole thing is the derivative of e^x^2."},{"Start":"04:58.175 ","End":"05:06.415","Text":"The integral is just e^x^2 evaluated from 0 to Pi and the answer is e^Pi^2 minus 1."},{"Start":"05:06.415 ","End":"05:10.650","Text":"That concludes this exercise and also this clip."}],"ID":28712}],"Thumbnail":null,"ID":294449},{"Name":"Real Fourier Series","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Simple Introduction","Duration":"6m 34s","ChapterTopicVideoID":27528,"CourseChapterTopicPlaylistID":294450,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.040","Text":"In this clip, we\u0027ll do a brief introduction informal to"},{"Start":"00:05.040 ","End":"00:07.830","Text":"the Fourier series of real functions because there\u0027s"},{"Start":"00:07.830 ","End":"00:11.340","Text":"also Fourier series for complex functions."},{"Start":"00:11.340 ","End":"00:13.830","Text":"We\u0027ll start with an example."},{"Start":"00:13.830 ","End":"00:16.770","Text":"Most basic thing is a 2Pi periodic function,"},{"Start":"00:16.770 ","End":"00:18.570","Text":"though it could have any period."},{"Start":"00:18.570 ","End":"00:22.620","Text":"We will take the example where on 0 to 2Pi is"},{"Start":"00:22.620 ","End":"00:28.785","Text":"defined to be equal to 1 from 0 to Pi and 0 from Pi to 2Pi."},{"Start":"00:28.785 ","End":"00:32.035","Text":"Here it\u0027s 1 and here it\u0027s 0."},{"Start":"00:32.035 ","End":"00:37.045","Text":"Sometimes we define it from minus Pi to Pi because that\u0027s symmetrical."},{"Start":"00:37.045 ","End":"00:42.655","Text":"We\u0027d say that here it\u0027s 0 and here it\u0027s 1."},{"Start":"00:42.655 ","End":"00:45.200","Text":"Once we have it on a 2Pi interval,"},{"Start":"00:45.200 ","End":"00:48.455","Text":"then we can extend it periodically to all of the reals."},{"Start":"00:48.455 ","End":"00:51.155","Text":"This case is what we call a square wave."},{"Start":"00:51.155 ","End":"00:56.330","Text":"If you want to approximate this function by some smooth series,"},{"Start":"00:56.330 ","End":"01:02.900","Text":"we can\u0027t do it with Taylor series not around x=0 because it\u0027s not continuous."},{"Start":"01:02.900 ","End":"01:04.580","Text":"We draw on a continuous line,"},{"Start":"01:04.580 ","End":"01:06.500","Text":"but this vertical line doesn\u0027t exist."},{"Start":"01:06.500 ","End":"01:09.355","Text":"Here it\u0027s 0 and here it\u0027s 1,"},{"Start":"01:09.355 ","End":"01:14.415","Text":"not continuous so it\u0027s not differentiable so we can\u0027t talk about a Taylor series,"},{"Start":"01:14.415 ","End":"01:16.890","Text":"Maclaurin series, it\u0027s also called."},{"Start":"01:16.890 ","End":"01:19.799","Text":"But we could try something else instead of a Taylor series."},{"Start":"01:19.799 ","End":"01:22.535","Text":"We could try other building blocks."},{"Start":"01:22.535 ","End":"01:25.925","Text":"For example, the following functions have a period of 2Pi,"},{"Start":"01:25.925 ","End":"01:30.410","Text":"a constant function which you could write as a constant times cosine 0x,"},{"Start":"01:30.410 ","End":"01:34.325","Text":"also cosine x, also sine x."},{"Start":"01:34.325 ","End":"01:41.315","Text":"In fact, cosine of any multiple of x will have a period of 2Pi."},{"Start":"01:41.315 ","End":"01:43.940","Text":"In these cases, also a smaller period,"},{"Start":"01:43.940 ","End":"01:47.090","Text":"but in particular it\u0027s periodic 2Pi. That\u0027s not all."},{"Start":"01:47.090 ","End":"01:49.790","Text":"If we take a linear combination of these,"},{"Start":"01:49.790 ","End":"01:52.100","Text":"then it will also be periodic with"},{"Start":"01:52.100 ","End":"01:55.945","Text":"2Pi and we\u0027ll even take an infinite linear combination."},{"Start":"01:55.945 ","End":"01:59.210","Text":"That\u0027s a symbol like so the tilde,"},{"Start":"01:59.210 ","End":"02:04.010","Text":"which means that this is the Fourier series for this, we\u0027ll revisit that."},{"Start":"02:04.010 ","End":"02:06.650","Text":"Now a bit of background on the Fourier series."},{"Start":"02:06.650 ","End":"02:13.045","Text":"Well, the person called Joseph Fourier or his full name and here\u0027s a picture."},{"Start":"02:13.045 ","End":"02:19.070","Text":"He started to investigate such theories trying to solve the heat equation."},{"Start":"02:19.070 ","End":"02:21.485","Text":"There\u0027s a thing called the heat equation,"},{"Start":"02:21.485 ","End":"02:25.444","Text":"which talks about heat conduction in a rod,"},{"Start":"02:25.444 ","End":"02:32.990","Text":"where it turns out that the change of temperature at a point over time is proportional"},{"Start":"02:32.990 ","End":"02:36.080","Text":"to the second derivative or the rate of change of"},{"Start":"02:36.080 ","End":"02:40.755","Text":"the gradient of temperature over the length of the rod."},{"Start":"02:40.755 ","End":"02:43.335","Text":"It\u0027s a function of x and time."},{"Start":"02:43.335 ","End":"02:48.315","Text":"The more generally it will be 3 dimensions of space and 1 dimension of time."},{"Start":"02:48.315 ","End":"02:54.500","Text":"You have space variable and a time variable and we take it over a finite length of rod."},{"Start":"02:54.500 ","End":"02:56.638","Text":"There\u0027s an initial condition."},{"Start":"02:56.638 ","End":"02:59.120","Text":"We\u0027re given the value of the temperature,"},{"Start":"02:59.120 ","End":"03:02.270","Text":"the heat at time 0 over the rod."},{"Start":"03:02.270 ","End":"03:07.220","Text":"Also that the temperature here and here is 0,"},{"Start":"03:07.220 ","End":"03:09.355","Text":"whatever the time is."},{"Start":"03:09.355 ","End":"03:11.948","Text":"Just as I say, historical background."},{"Start":"03:11.948 ","End":"03:18.485","Text":"You don\u0027t have to study this, and he found that if you take the function sine x,"},{"Start":"03:18.485 ","End":"03:23.660","Text":"or more generally sine of something times x and you could find a solution to the equation"},{"Start":"03:23.660 ","End":"03:29.120","Text":"as u(x,t)=sine of Pi x over L times e to the,"},{"Start":"03:29.120 ","End":"03:30.905","Text":"what it says here."},{"Start":"03:30.905 ","End":"03:38.150","Text":"It turns out this has period L. The Pi over L makes it change from period Pi to period"},{"Start":"03:38.150 ","End":"03:45.305","Text":"L. If you check this derivative with respect to x would be first of all,"},{"Start":"03:45.305 ","End":"03:48.635","Text":"Pi over L cosine time of this,"},{"Start":"03:48.635 ","End":"03:53.100","Text":"and then minus Pi^2 over L^2 times sine of this."},{"Start":"03:53.100 ","End":"03:55.790","Text":"If you differentiate it with respect to t,"},{"Start":"03:55.790 ","End":"03:58.880","Text":"It\u0027s k times minus pi^2 over L^2."},{"Start":"03:58.880 ","End":"04:03.070","Text":"Anyway, it solved this equation and the initial conditions."},{"Start":"04:03.070 ","End":"04:10.925","Text":"If we take f(x) to be sine of Pi x over L. Then when he found these building blocks,"},{"Start":"04:10.925 ","End":"04:14.420","Text":"found that if you can just sum them as a series,"},{"Start":"04:14.420 ","End":"04:17.145","Text":"then it\u0027s also a solution."},{"Start":"04:17.145 ","End":"04:22.235","Text":"There\u0027s a formula for this coefficient and this is not important."},{"Start":"04:22.235 ","End":"04:26.390","Text":"But the point here is these Fourier series with the sign and"},{"Start":"04:26.390 ","End":"04:30.935","Text":"later sine and cosine started out with trying to solve the heat equation."},{"Start":"04:30.935 ","End":"04:33.500","Text":"Nowadays, Fourier series have lots of uses in"},{"Start":"04:33.500 ","End":"04:38.075","Text":"mathematics and engineering and especially electrical engineering."},{"Start":"04:38.075 ","End":"04:42.185","Text":"I\u0027ll show you now the notation for Fourier series."},{"Start":"04:42.185 ","End":"04:44.855","Text":"Don\u0027t worry exactly what this Tilde means."},{"Start":"04:44.855 ","End":"04:48.140","Text":"Fourier series for f is this."},{"Start":"04:48.140 ","End":"04:50.450","Text":"Also instead of writing it with a dot, dot,"},{"Start":"04:50.450 ","End":"04:53.045","Text":"dot, we use the summation."},{"Start":"04:53.045 ","End":"04:55.850","Text":"Every alternate term is a cosine."},{"Start":"04:55.850 ","End":"05:03.475","Text":"We have the infinite sum of something times cosine nx plus something times sine nx."},{"Start":"05:03.475 ","End":"05:08.990","Text":"If we just take the sum instead of to infinity to some number n,"},{"Start":"05:08.990 ","End":"05:16.305","Text":"then we get what is called the nth approximation to the function f."},{"Start":"05:16.305 ","End":"05:24.205","Text":"There\u0027s a formula for the coefficients a_n is the inner product of f with cosine nx,"},{"Start":"05:24.205 ","End":"05:26.730","Text":"is the inner product of the L2 space,"},{"Start":"05:26.730 ","End":"05:28.490","Text":"and it\u0027s equal to this."},{"Start":"05:28.490 ","End":"05:30.380","Text":"If you plug in n=0,"},{"Start":"05:30.380 ","End":"05:33.065","Text":"you get the coefficients a naught."},{"Start":"05:33.065 ","End":"05:35.695","Text":"There is also a formula for the b_n."},{"Start":"05:35.695 ","End":"05:40.535","Text":"I said, this is the inner product on this space L2,"},{"Start":"05:40.535 ","End":"05:44.045","Text":"piecewise continuous on the interval minus Pi to Pi,"},{"Start":"05:44.045 ","End":"05:48.457","Text":"which you could extend periodically to the whole number line."},{"Start":"05:48.457 ","End":"05:51.950","Text":"An example, we\u0027ll take the previous example we"},{"Start":"05:51.950 ","End":"05:56.030","Text":"had of the square wave function which on 0-2pi,"},{"Start":"05:56.030 ","End":"05:57.770","Text":"it\u0027s defined as follows."},{"Start":"05:57.770 ","End":"05:59.600","Text":"Yeah, here\u0027s the picture."},{"Start":"05:59.600 ","End":"06:05.350","Text":"It turns out that if you compute the second approximation S_2,"},{"Start":"06:05.350 ","End":"06:07.964","Text":"go back and see how we defined S_n,"},{"Start":"06:07.964 ","End":"06:09.295","Text":"it looks like this,"},{"Start":"06:09.295 ","End":"06:12.260","Text":"which is not a great approximation,"},{"Start":"06:12.260 ","End":"06:14.210","Text":"but as n gets bigger here,"},{"Start":"06:14.210 ","End":"06:16.865","Text":"it gets closer and closer."},{"Start":"06:16.865 ","End":"06:20.720","Text":"I\u0027ll show you a little bit of videos so you can see what I mean,"},{"Start":"06:20.720 ","End":"06:24.060","Text":"and with the video, we\u0027ll end this clip."}],"ID":28713},{"Watched":false,"Name":"Example","Duration":"2m 33s","ChapterTopicVideoID":27529,"CourseChapterTopicPlaylistID":294450,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.185","Text":"Now let\u0027s do an actual example of finding the Fourier series of a real function."},{"Start":"00:07.185 ","End":"00:15.585","Text":"We\u0027ll take the function as f(x)=x on the interval from minus Pi to Pi."},{"Start":"00:15.585 ","End":"00:17.850","Text":"This is what the graph looks like."},{"Start":"00:17.850 ","End":"00:21.690","Text":"It\u0027s just this piece from minus Pi to Pi but if you extend it periodically,"},{"Start":"00:21.690 ","End":"00:23.790","Text":"you get an infinite number of these."},{"Start":"00:23.790 ","End":"00:27.000","Text":"Anyway by the formula a_n,"},{"Start":"00:27.000 ","End":"00:30.060","Text":"you can remember it as the inner product of f with cosine nx,"},{"Start":"00:30.060 ","End":"00:34.420","Text":"or you can just straight away go to the formula with the integral."},{"Start":"00:34.420 ","End":"00:36.345","Text":"If we compute this,"},{"Start":"00:36.345 ","End":"00:38.385","Text":"because this is an odd function,"},{"Start":"00:38.385 ","End":"00:43.985","Text":"because it\u0027s an odd times an even function on a symmetric interval this is equal to 0."},{"Start":"00:43.985 ","End":"00:51.335","Text":"The b_n, using the formula with sine instead of cosine comes out to be this,"},{"Start":"00:51.335 ","End":"00:54.380","Text":"which we do with integration by parts."},{"Start":"00:54.380 ","End":"00:59.600","Text":"This is the formula for integration by parts and won\u0027t go into all the details."},{"Start":"00:59.600 ","End":"01:01.010","Text":"You know how to do this."},{"Start":"01:01.010 ","End":"01:06.150","Text":"The answer comes out to be, finally,"},{"Start":"01:06.150 ","End":"01:11.480","Text":"minus 1^n plus 1 times 2 over n. It alternates plus or"},{"Start":"01:11.480 ","End":"01:17.765","Text":"minus with 2 over n. Now if we plug a_n and b_n back in the formula,"},{"Start":"01:17.765 ","End":"01:20.460","Text":"then we get the following."},{"Start":"01:20.460 ","End":"01:25.390","Text":"We don\u0027t really need to write these because these a\u0027s are 0,"},{"Start":"01:25.390 ","End":"01:28.190","Text":"so it\u0027s just the following,"},{"Start":"01:28.190 ","End":"01:31.415","Text":"and that\u0027s our Fourier series for the function x."},{"Start":"01:31.415 ","End":"01:35.465","Text":"If we want the approximation and instead of taking infinity here,"},{"Start":"01:35.465 ","End":"01:40.115","Text":"we just take it up to n and we get S_N(x)."},{"Start":"01:40.115 ","End":"01:46.485","Text":"For example, S_2(x) comes out to be the sum from 1-2."},{"Start":"01:46.485 ","End":"01:51.825","Text":"If n=1, then this is a plus 2 over 1 sine of 1x."},{"Start":"01:51.825 ","End":"01:54.250","Text":"It\u0027s this and if n is 2,"},{"Start":"01:54.250 ","End":"01:57.860","Text":"then we get a minus and we get 2 over 2,"},{"Start":"01:57.860 ","End":"02:03.565","Text":"which is 1 sine of 2x and this looks like this."},{"Start":"02:03.565 ","End":"02:07.150","Text":"Although it\u0027s not a great approximation, that\u0027s only S_2."},{"Start":"02:07.150 ","End":"02:10.130","Text":"If we let n get bigger and bigger and go to infinity,"},{"Start":"02:10.130 ","End":"02:12.425","Text":"it will get closer and closer."},{"Start":"02:12.425 ","End":"02:17.320","Text":"I\u0027ll end this clip with a video of that."}],"ID":28714},{"Watched":false,"Name":"Advanced Introduction","Duration":"4m 6s","ChapterTopicVideoID":27527,"CourseChapterTopicPlaylistID":294450,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.290","Text":"We already had an intro to Fourier series of real functions that was a basic intro."},{"Start":"00:06.290 ","End":"00:08.115","Text":"This is a bit more advanced."},{"Start":"00:08.115 ","End":"00:09.930","Text":"We\u0027ll be concerned with functions,"},{"Start":"00:09.930 ","End":"00:16.320","Text":"but these functions belong to an inner product space called L^2_PC."},{"Start":"00:16.320 ","End":"00:19.110","Text":"The PC is for piecewise continuous."},{"Start":"00:19.110 ","End":"00:23.939","Text":"The L^2 is because of the way we define the norm and the inner product."},{"Start":"00:23.939 ","End":"00:26.580","Text":"But it\u0027s also called for short E. Now,"},{"Start":"00:26.580 ","End":"00:29.580","Text":"we\u0027ll be studying it on the interval from minus Pi to Pi,"},{"Start":"00:29.580 ","End":"00:31.675","Text":"which is the most natural."},{"Start":"00:31.675 ","End":"00:34.370","Text":"But in a later chapter section,"},{"Start":"00:34.370 ","End":"00:38.690","Text":"we\u0027ll talk about generalizing this to a general interval a, b."},{"Start":"00:38.690 ","End":"00:45.000","Text":"For short, we use a single letter E instead of L^2_PC, it\u0027s easier."},{"Start":"00:45.000 ","End":"00:50.120","Text":"Piecewise continuous functions on the interval from minus Pi to Pi."},{"Start":"00:50.120 ","End":"00:52.310","Text":"Now I said inner product space."},{"Start":"00:52.310 ","End":"00:57.635","Text":"The addition and subtraction and scalar multiplication are clear for functions here."},{"Start":"00:57.635 ","End":"00:58.895","Text":"It\u0027s a vector space."},{"Start":"00:58.895 ","End":"01:00.230","Text":"What about the inner product?"},{"Start":"01:00.230 ","End":"01:01.760","Text":"We\u0027ll be defined it as follows."},{"Start":"01:01.760 ","End":"01:05.270","Text":"The product of f with g inner product, dot-product,"},{"Start":"01:05.270 ","End":"01:13.030","Text":"scalar product is 1 over Pi integral from minus Pi to Pi of f times gdx."},{"Start":"01:13.030 ","End":"01:15.680","Text":"Sometimes we write a conjugate sign over"},{"Start":"01:15.680 ","End":"01:18.259","Text":"here because that\u0027s what it is with complex functions."},{"Start":"01:18.259 ","End":"01:19.775","Text":"But if real functions,"},{"Start":"01:19.775 ","End":"01:21.950","Text":"the conjugate is the same as the thing itself."},{"Start":"01:21.950 ","End":"01:24.530","Text":"I\u0027ve omitted the bar over the g(x)."},{"Start":"01:24.530 ","End":"01:26.420","Text":"Sometimes you will see it."},{"Start":"01:26.420 ","End":"01:31.985","Text":"This space with this product has an infinite orthonormal system."},{"Start":"01:31.985 ","End":"01:33.995","Text":"It consists of a single function,"},{"Start":"01:33.995 ","End":"01:36.305","Text":"1 over square root of 2 constant,"},{"Start":"01:36.305 ","End":"01:38.210","Text":"and family of functions,"},{"Start":"01:38.210 ","End":"01:39.620","Text":"all the sine nx,"},{"Start":"01:39.620 ","End":"01:42.530","Text":"where n is from 1 to infinity and the cosine nx,"},{"Start":"01:42.530 ","End":"01:44.450","Text":"and goes from 1 to infinity."},{"Start":"01:44.450 ","End":"01:47.795","Text":"This is strictly speaking up correct mathematical notation"},{"Start":"01:47.795 ","End":"01:50.000","Text":"should really write it as a union,"},{"Start":"01:50.000 ","End":"01:53.150","Text":"but there\u0027s no confusion in writing it this way."},{"Start":"01:53.150 ","End":"01:58.280","Text":"Now, Fourier series are a way of approximating functions from"},{"Start":"01:58.280 ","End":"02:03.650","Text":"this space with trigonometric functions like sine and cosine."},{"Start":"02:03.650 ","End":"02:08.630","Text":"Just like the polynomials where a Taylor series and Maclaurin"},{"Start":"02:08.630 ","End":"02:13.475","Text":"series for functions that have derivatives of every order at a point."},{"Start":"02:13.475 ","End":"02:15.260","Text":"We get Taylor series."},{"Start":"02:15.260 ","End":"02:17.120","Text":"Well, with these functions,"},{"Start":"02:17.120 ","End":"02:19.910","Text":"we can\u0027t get Taylor series because there may not be"},{"Start":"02:19.910 ","End":"02:22.895","Text":"continuous even if they were differentiable,"},{"Start":"02:22.895 ","End":"02:26.230","Text":"but we will get Fourier series and we will see this."},{"Start":"02:26.230 ","End":"02:32.000","Text":"The real Fourier series of a function f is defined to be the following sum."},{"Start":"02:32.000 ","End":"02:35.045","Text":"I\u0027m not claiming there\u0027s inequality."},{"Start":"02:35.045 ","End":"02:38.090","Text":"This is the representation of f as"},{"Start":"02:38.090 ","End":"02:41.690","Text":"a Fourier series and we just write this till, this wave."},{"Start":"02:41.690 ","End":"02:44.870","Text":"We\u0027ll see to what extent this really does equal"},{"Start":"02:44.870 ","End":"02:48.245","Text":"this later or how does this approximate this."},{"Start":"02:48.245 ","End":"02:50.375","Text":"It\u0027s a_naught over 2."},{"Start":"02:50.375 ","End":"02:52.870","Text":"There\u0027s a reason for taking it over 2."},{"Start":"02:52.870 ","End":"02:59.940","Text":"Then the sum of coefficients a_n times cosine nx plus sum of b_n sine nx."},{"Start":"02:59.940 ","End":"03:02.135","Text":"The sum goes from 1 to infinity."},{"Start":"03:02.135 ","End":"03:06.815","Text":"There is a formula for the coefficients a_n and b_n."},{"Start":"03:06.815 ","End":"03:12.575","Text":"A_n is equal to the inner product of f with cosine nx,"},{"Start":"03:12.575 ","End":"03:14.915","Text":"and it\u0027s given by this integral."},{"Start":"03:14.915 ","End":"03:18.390","Text":"That\u0027s true for n=0,"},{"Start":"03:18.390 ","End":"03:20.160","Text":"1, 2, 3, 4, 5."},{"Start":"03:20.160 ","End":"03:22.215","Text":"Also good for n=0."},{"Start":"03:22.215 ","End":"03:27.210","Text":"Then 0, cosine nx is just the constant function 1."},{"Start":"03:27.210 ","End":"03:33.470","Text":"Usually, we give a separate formula that a_naught if you let the cosine drop off here,"},{"Start":"03:33.470 ","End":"03:35.495","Text":"when any 0s there is no cosine."},{"Start":"03:35.495 ","End":"03:37.820","Text":"It\u0027s just the integral of f(x)dx."},{"Start":"03:37.820 ","End":"03:39.590","Text":"The formula for b_n is similar."},{"Start":"03:39.590 ","End":"03:41.540","Text":"Instead of taking the inner product with the cosine,"},{"Start":"03:41.540 ","End":"03:43.850","Text":"we take an inner product with the sine function."},{"Start":"03:43.850 ","End":"03:45.935","Text":"And this is what it is at an integral."},{"Start":"03:45.935 ","End":"03:47.720","Text":"Well, that\u0027s it in a nutshell."},{"Start":"03:47.720 ","End":"03:53.750","Text":"The exercises will typically consist of given a function on minus Pi Pi,"},{"Start":"03:53.750 ","End":"03:57.650","Text":"we have to figure out the coefficients a_n and b_n and then"},{"Start":"03:57.650 ","End":"04:03.130","Text":"write our function f as a Fourier series like so."},{"Start":"04:03.130 ","End":"04:06.460","Text":"That\u0027s all for this introduction."}],"ID":28715},{"Watched":false,"Name":"Exercise 1","Duration":"3m 9s","ChapterTopicVideoID":27530,"CourseChapterTopicPlaylistID":294450,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"In this exercise, we\u0027re going to compute"},{"Start":"00:02.550 ","End":"00:07.845","Text":"the real Fourier series for the function f(x) on the interval from minus Pi to Pi,"},{"Start":"00:07.845 ","End":"00:11.025","Text":"where f is defined piecewise as here."},{"Start":"00:11.025 ","End":"00:14.220","Text":"It\u0027s 0 when x is negative 1 or x is positive,"},{"Start":"00:14.220 ","End":"00:16.155","Text":"and here\u0027s a little picture."},{"Start":"00:16.155 ","End":"00:20.585","Text":"The formula for the Fourier series is this one."},{"Start":"00:20.585 ","End":"00:23.585","Text":"We\u0027re going to compute the a_n and the b_n."},{"Start":"00:23.585 ","End":"00:27.920","Text":"a_n is given by this formula and in our case,"},{"Start":"00:27.920 ","End":"00:32.360","Text":"we just have to take the part from 0 to Pi where it\u0027s"},{"Start":"00:32.360 ","End":"00:37.345","Text":"1 and we don\u0027t need f(x) will replace that by 1 from 0 to Pi."},{"Start":"00:37.345 ","End":"00:40.505","Text":"This is the integral we get."},{"Start":"00:40.505 ","End":"00:44.490","Text":"This comes out to be the integral of"},{"Start":"00:44.490 ","End":"00:50.045","Text":"cosine nx is sine nx over n. We take it from 0 to Pi."},{"Start":"00:50.045 ","End":"00:52.750","Text":"Of course this only applies when n is not equal to 0."},{"Start":"00:52.750 ","End":"00:56.165","Text":"We\u0027ll have to compute the case n=0 separately."},{"Start":"00:56.165 ","End":"00:59.025","Text":"This is equal to 0,"},{"Start":"00:59.025 ","End":"01:01.480","Text":"and there\u0027s 0 is the following formula."},{"Start":"01:01.480 ","End":"01:06.670","Text":"You can always use this one again with cosine of 0x being 1."},{"Start":"01:06.670 ","End":"01:10.995","Text":"We get just the integral of 1dx 0 to Pi."},{"Start":"01:10.995 ","End":"01:13.350","Text":"It\u0027s Pi over Pi is 1."},{"Start":"01:13.350 ","End":"01:15.225","Text":"Now we\u0027ll continue on to b_n."},{"Start":"01:15.225 ","End":"01:20.250","Text":"b_n was similar to a_n except with sine instead of cosine."},{"Start":"01:20.250 ","End":"01:25.470","Text":"Again, we get integral from 0 to Pi(1),"},{"Start":"01:25.470 ","End":"01:29.920","Text":"and this comes out to be the integral of sine is minus cosine."},{"Start":"01:29.920 ","End":"01:34.330","Text":"Similar to above, which substitute the limits."},{"Start":"01:34.330 ","End":"01:39.620","Text":"We switched the upper and lower around and get rid of the minus makes it easier."},{"Start":"01:39.620 ","End":"01:41.475","Text":"We get this,"},{"Start":"01:41.475 ","End":"01:49.280","Text":"which is equal to cosine of n Pi is minus 1_n."},{"Start":"01:49.280 ","End":"01:51.560","Text":"We get this formula."},{"Start":"01:51.560 ","End":"01:58.265","Text":"Note that this will either be 0 or 2 depending on whether n is odd or even."},{"Start":"01:58.265 ","End":"02:01.940","Text":"We can put that in the table and see that if n is even,"},{"Start":"02:01.940 ","End":"02:06.785","Text":"that n is 2k, then this is 1,1 minus 1 is 0."},{"Start":"02:06.785 ","End":"02:09.995","Text":"If n is odd, 2k minus 1,"},{"Start":"02:09.995 ","End":"02:12.680","Text":"then this is equal to minus 1,"},{"Start":"02:12.680 ","End":"02:18.980","Text":"so 1 minus minus 1 is 2 and n is then 2k minus 1."},{"Start":"02:18.980 ","End":"02:21.190","Text":"Now let\u0027s put all this together."},{"Start":"02:21.190 ","End":"02:23.985","Text":"This is what we found for b_n."},{"Start":"02:23.985 ","End":"02:25.710","Text":"This is what a_n was."},{"Start":"02:25.710 ","End":"02:32.620","Text":"It was equal to 0 for most n except for n=0 where it was equal to 1."},{"Start":"02:32.620 ","End":"02:35.440","Text":"Now if we put everything in this formula,"},{"Start":"02:35.440 ","End":"02:40.210","Text":"then this part is going to be 0 because that\u0027s where n is bigger or equal to 1."},{"Start":"02:40.210 ","End":"02:42.115","Text":"We\u0027ll just get this and this."},{"Start":"02:42.115 ","End":"02:44.590","Text":"What we\u0027ll get is a naught over 2,"},{"Start":"02:44.590 ","End":"02:49.445","Text":"which is 1 over 2 plus the b_n, which is this."},{"Start":"02:49.445 ","End":"02:53.830","Text":"We just take all the 2k minus 1 from k equals 1 to infinity."},{"Start":"02:53.830 ","End":"02:55.407","Text":"This thing will go 1,"},{"Start":"02:55.407 ","End":"02:57.250","Text":"3, 5, 7, etc."},{"Start":"02:57.250 ","End":"03:02.210","Text":"Similarly, the n in front of the x gets replaced by 2k minus 1."},{"Start":"03:02.210 ","End":"03:04.260","Text":"This is the answer."},{"Start":"03:04.260 ","End":"03:06.735","Text":"Perhaps I\u0027ll highlight it,"},{"Start":"03:06.735 ","End":"03:10.090","Text":"and we are done."}],"ID":28716},{"Watched":false,"Name":"Exercise 2","Duration":"6m 35s","ChapterTopicVideoID":27531,"CourseChapterTopicPlaylistID":294450,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we\u0027re going to compute the real Fourier series for the function"},{"Start":"00:04.320 ","End":"00:08.475","Text":"f(x) on the interval minus Pi to Pi,"},{"Start":"00:08.475 ","End":"00:12.705","Text":"where f(x) is the sine of the absolute value of x."},{"Start":"00:12.705 ","End":"00:16.465","Text":"What does it look like? Here it is."},{"Start":"00:16.465 ","End":"00:19.905","Text":"From here to here, it\u0027s like sine x and then we reflect it,"},{"Start":"00:19.905 ","End":"00:21.944","Text":"make it an even function."},{"Start":"00:21.944 ","End":"00:27.240","Text":"If absolute value of x is an even function then any function of an even function is even."},{"Start":"00:27.240 ","End":"00:29.850","Text":"It doesn\u0027t matter if it\u0027s sine, cosine, or anything."},{"Start":"00:29.850 ","End":"00:34.230","Text":"This is a general formula for the Fourier series in the real case."},{"Start":"00:34.230 ","End":"00:37.420","Text":"We have formulas for a_n and b_n,"},{"Start":"00:37.420 ","End":"00:40.725","Text":"a_n is this, let\u0027s compute it."},{"Start":"00:40.725 ","End":"00:42.735","Text":"This is equal to,"},{"Start":"00:42.735 ","End":"00:46.010","Text":"instead of f(x) sine of absolute value of x."},{"Start":"00:46.010 ","End":"00:54.796","Text":"Because f(x) or sine absolute value of x is even and cosine is even,"},{"Start":"00:54.796 ","End":"00:57.890","Text":"altogether we have an even function."},{"Start":"00:57.890 ","End":"00:59.630","Text":"If it\u0027s an even function,"},{"Start":"00:59.630 ","End":"01:02.270","Text":"we can take twice the integral from 0 to Pi"},{"Start":"01:02.270 ","End":"01:05.179","Text":"because this integral is going to be the same as this integral."},{"Start":"01:05.179 ","End":"01:10.310","Text":"Yeah, so 2 over Pi and then just from 0 to Pi."},{"Start":"01:10.310 ","End":"01:13.160","Text":"Of course here in this part,"},{"Start":"01:13.160 ","End":"01:15.575","Text":"the absolute value of x is just x."},{"Start":"01:15.575 ","End":"01:21.500","Text":"This is equal to using this trigonometric formula for sine Alpha cosine Beta."},{"Start":"01:21.500 ","End":"01:24.590","Text":"Plug it in here and we get this."},{"Start":"01:24.590 ","End":"01:30.400","Text":"The 2 here, swallowed up the 2 here in case you\u0027re wondering."},{"Start":"01:30.400 ","End":"01:33.065","Text":"Let\u0027s do this integral."},{"Start":"01:33.065 ","End":"01:39.320","Text":"This is equal to 1 over Pi and then this is going to be minus cosine and we"},{"Start":"01:39.320 ","End":"01:45.645","Text":"also have to divide by 1 minus n. Here we have to divide by 1 plus n. Because of this,"},{"Start":"01:45.645 ","End":"01:47.665","Text":"n can\u0027t equal 1."},{"Start":"01:47.665 ","End":"01:49.760","Text":"Sorry, there\u0027s a blank line here."},{"Start":"01:49.760 ","End":"01:51.695","Text":"Now this is equal to."},{"Start":"01:51.695 ","End":"01:55.520","Text":"Switch the limits of substitution so"},{"Start":"01:55.520 ","End":"01:59.460","Text":"that\u0027s going to be 0 at the top and Pi at the bottom,"},{"Start":"01:59.460 ","End":"02:05.430","Text":"and this minus disappears and so does this minus."},{"Start":"02:05.430 ","End":"02:09.160","Text":"Now, when x is 0,"},{"Start":"02:09.160 ","End":"02:11.470","Text":"that\u0027s cosine of 0."},{"Start":"02:11.470 ","End":"02:12.730","Text":"This is cosine of 0."},{"Start":"02:12.730 ","End":"02:14.380","Text":"That\u0027s 1, that\u0027s 1."},{"Start":"02:14.380 ","End":"02:16.510","Text":"When we plug in Pi,"},{"Start":"02:16.510 ","End":"02:19.090","Text":"we\u0027ve got cosine 1 minus nPi,"},{"Start":"02:19.090 ","End":"02:21.340","Text":"but I prefer to write it as n minus 1,"},{"Start":"02:21.340 ","End":"02:24.260","Text":"since cosine is even, it doesn\u0027t matter."},{"Start":"02:24.260 ","End":"02:27.160","Text":"Here, cosine n plus 1 Pi."},{"Start":"02:27.160 ","End":"02:29.980","Text":"These are both to the minus because we\u0027re subtracting."},{"Start":"02:29.980 ","End":"02:32.150","Text":"This comes out to be,"},{"Start":"02:32.150 ","End":"02:34.000","Text":"this part\u0027s the same."},{"Start":"02:34.000 ","End":"02:38.095","Text":"Cosine of mPi is minus 1^m."},{"Start":"02:38.095 ","End":"02:42.280","Text":"Here take m as n minus 1 and we have minus 1^n minus 1"},{"Start":"02:42.280 ","End":"02:47.060","Text":"here minus 1^n plus 1 and everything else is the same."},{"Start":"02:47.060 ","End":"02:53.755","Text":"Now, this part we\u0027re going to add together in a moment and we have this also over here,"},{"Start":"02:53.755 ","End":"02:57.100","Text":"but with a minus 1^n prefix."},{"Start":"02:57.100 ","End":"02:59.785","Text":"Now, what did this minus 1^n come from?"},{"Start":"02:59.785 ","End":"03:02.490","Text":"If we take minus 1^n plus 1,"},{"Start":"03:02.490 ","End":"03:07.755","Text":"that\u0027s equal to minus 1^2 minus 1^n minus 1 and this is 1."},{"Start":"03:07.755 ","End":"03:11.410","Text":"The minus 1^n plus 1 equals minus 1^n minus 1."},{"Start":"03:11.410 ","End":"03:17.920","Text":"These 2 are equal and they\u0027re both equal to minus 1^n plus 1."},{"Start":"03:17.920 ","End":"03:22.060","Text":"We can take one of the minuses out here and that cancels with the minus here,"},{"Start":"03:22.060 ","End":"03:25.480","Text":"it makes it a plus, so it\u0027s just minus 1^n."},{"Start":"03:25.480 ","End":"03:32.280","Text":"Again we have just 1 over 1 minus n plus 1 over 1 plus n both here and here."},{"Start":"03:32.280 ","End":"03:34.985","Text":"That comes out where you can mentally do it."},{"Start":"03:34.985 ","End":"03:39.080","Text":"We cross multiply 1 plus n and 1 minus"},{"Start":"03:39.080 ","End":"03:44.614","Text":"n is just 2 because the n cancels and this time this is 1 minus n^2."},{"Start":"03:44.614 ","End":"03:46.925","Text":"We have one time from here,"},{"Start":"03:46.925 ","End":"03:48.950","Text":"minus 1^n times from here,"},{"Start":"03:48.950 ","End":"03:51.845","Text":"so this is what we get and there is the Pi on the bottom."},{"Start":"03:51.845 ","End":"03:55.415","Text":"Now if we separate this to n odd or even,"},{"Start":"03:55.415 ","End":"03:57.990","Text":"if n is even,"},{"Start":"03:57.990 ","End":"04:00.940","Text":"then minus 1^n comes out to be 1,"},{"Start":"04:00.940 ","End":"04:03.645","Text":"1 plus 1 is 2 times 2 is 4."},{"Start":"04:03.645 ","End":"04:10.275","Text":"We get 4 over Pi and then 1 over 1 minus n squared but n is 2k."},{"Start":"04:10.275 ","End":"04:13.855","Text":"If n is odd,"},{"Start":"04:13.855 ","End":"04:19.835","Text":"then this thing comes out to be minus 1 so 1 and minus 1 is 0,"},{"Start":"04:19.835 ","End":"04:25.010","Text":"but we still have to exclude n=1 like we previously had to exclude it."},{"Start":"04:25.010 ","End":"04:27.380","Text":"Now we\u0027re going to compute a_1."},{"Start":"04:27.380 ","End":"04:30.640","Text":"Remember this is the formula for a_n."},{"Start":"04:30.640 ","End":"04:35.490","Text":"So a_1 is what we get if we put n=1, n is 1,"},{"Start":"04:35.490 ","End":"04:36.840","Text":"x minus x is 0,"},{"Start":"04:36.840 ","End":"04:40.395","Text":"sine of 0 is 0, so this term drops out."},{"Start":"04:40.395 ","End":"04:43.650","Text":"Here we have x plus 1x is 2x,"},{"Start":"04:43.650 ","End":"04:46.860","Text":"so we just have sine of 2x dx."},{"Start":"04:46.860 ","End":"04:49.980","Text":"The integral of sine 2x is minus cosine 2x,"},{"Start":"04:49.980 ","End":"04:53.270","Text":"but divided by 2 from 0 to Pi."},{"Start":"04:53.270 ","End":"04:58.850","Text":"Get rid of the minus by reversing the order of 0 and Pi."},{"Start":"04:58.850 ","End":"05:02.825","Text":"This comes out to be cosine of 0 is 1,"},{"Start":"05:02.825 ","End":"05:05.945","Text":"cosine of 2Pi is also 1."},{"Start":"05:05.945 ","End":"05:08.150","Text":"This comes out to be 0."},{"Start":"05:08.150 ","End":"05:11.465","Text":"That means that our formula for a_n,"},{"Start":"05:11.465 ","End":"05:15.050","Text":"whereas previously we excluded n=1."},{"Start":"05:15.050 ","End":"05:18.845","Text":"Now we don\u0027t have to exclude it because when n=1, it\u0027s also 0."},{"Start":"05:18.845 ","End":"05:25.410","Text":"We just have a clean separation of even and odd."},{"Start":"05:25.550 ","End":"05:27.690","Text":"That\u0027s just a_n."},{"Start":"05:27.690 ","End":"05:29.805","Text":"Now we have to get round to b_n,"},{"Start":"05:29.805 ","End":"05:32.340","Text":"b_n is this integral."},{"Start":"05:32.340 ","End":"05:38.060","Text":"Absolute value of x is even so any function of an even function is also even."},{"Start":"05:38.060 ","End":"05:39.455","Text":"This sine is odd,"},{"Start":"05:39.455 ","End":"05:41.750","Text":"even times odd is odd."},{"Start":"05:41.750 ","End":"05:46.955","Text":"The integral of an odd function on a symmetric interval is just 0,"},{"Start":"05:46.955 ","End":"05:48.680","Text":"so b_n is 0."},{"Start":"05:48.680 ","End":"05:51.485","Text":"That came out nice. Let\u0027s see."},{"Start":"05:51.485 ","End":"05:55.430","Text":"Well the b_n\u0027s are 0 so in the Fourier series,"},{"Start":"05:55.430 ","End":"06:00.025","Text":"this whole b_n sine and x sum just disappears."},{"Start":"06:00.025 ","End":"06:03.045","Text":"We want a Naught over 2 and then this sum,"},{"Start":"06:03.045 ","End":"06:08.040","Text":"but not all n only the n that are equal to 2k."},{"Start":"06:08.040 ","End":"06:10.935","Text":"What we get is the following,"},{"Start":"06:10.935 ","End":"06:12.390","Text":"a Naught over 2,"},{"Start":"06:12.390 ","End":"06:15.065","Text":"well a Naught is 4 over Pi,"},{"Start":"06:15.065 ","End":"06:17.120","Text":"because when k is 0, this is 1,"},{"Start":"06:17.120 ","End":"06:18.860","Text":"so it\u0027s just 4 over Pi over 2,"},{"Start":"06:18.860 ","End":"06:20.555","Text":"which is 2 over Pi."},{"Start":"06:20.555 ","End":"06:21.920","Text":"Then we have this sum,"},{"Start":"06:21.920 ","End":"06:24.590","Text":"but instead of n going from 1 to infinity,"},{"Start":"06:24.590 ","End":"06:28.560","Text":"we have k going from 1 to infinity and n is 2k."},{"Start":"06:28.560 ","End":"06:35.080","Text":"There is a 2k here and a 2k here and that\u0027s our answer and we\u0027re done."}],"ID":28717},{"Watched":false,"Name":"Exercise 3","Duration":"2m 33s","ChapterTopicVideoID":27532,"CourseChapterTopicPlaylistID":294450,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.800","Text":"In this exercise, we\u0027re going to compute the Fourier series for the function f that\u0027s"},{"Start":"00:04.800 ","End":"00:09.585","Text":"given over here on the interval from minus Pi to Pi."},{"Start":"00:09.585 ","End":"00:15.810","Text":"F is 1 between minus 1 and 1 and 0 otherwise."},{"Start":"00:15.810 ","End":"00:21.480","Text":"Here\u0027s a picture, the graph of f. Anyway,"},{"Start":"00:21.480 ","End":"00:25.755","Text":"the formula for the Fourier series is this,"},{"Start":"00:25.755 ","End":"00:30.330","Text":"and we\u0027re given formula for each of the coefficients,"},{"Start":"00:30.330 ","End":"00:32.820","Text":"I mean for the a_n\u0027s and for the b_n\u0027s."},{"Start":"00:32.820 ","End":"00:35.610","Text":"The a_n is given by this formula."},{"Start":"00:35.610 ","End":"00:37.825","Text":"Let\u0027s compute the a_n\u0027s first."},{"Start":"00:37.825 ","End":"00:40.720","Text":"This is equal to instead of the integral from minus Pi to"},{"Start":"00:40.720 ","End":"00:43.985","Text":"Pi because it\u0027s 1 here and 0 otherwise,"},{"Start":"00:43.985 ","End":"00:50.990","Text":"we just change the limit to be from minus 1-1 because it\u0027s 0 outside that and 1 inside."},{"Start":"00:50.990 ","End":"00:53.355","Text":"Because it\u0027s an even function,"},{"Start":"00:53.355 ","End":"00:57.995","Text":"we can take the integral from 0-1 and double it."},{"Start":"00:57.995 ","End":"01:03.260","Text":"We get integral of cosine is sine but we have to divide by the integer"},{"Start":"01:03.260 ","End":"01:09.115","Text":"derivative n. We have 2 over Pi sine of x over n, 0-1."},{"Start":"01:09.115 ","End":"01:11.970","Text":"This formula doesn\u0027t work with n=0,"},{"Start":"01:11.970 ","End":"01:14.600","Text":"so we\u0027ll have to compute a_0 separately."},{"Start":"01:14.600 ","End":"01:16.710","Text":"Anyway, this comes out to be,"},{"Start":"01:16.710 ","End":"01:19.110","Text":"when x is 0 everything is 0."},{"Start":"01:19.110 ","End":"01:20.865","Text":"When x is 1,"},{"Start":"01:20.865 ","End":"01:23.880","Text":"we get sine n over n and the 2 over Pi."},{"Start":"01:23.880 ","End":"01:25.635","Text":"Because what we have for a_n,"},{"Start":"01:25.635 ","End":"01:30.320","Text":"we want to compute a_naught so we\u0027ll take the general formula for a_n and"},{"Start":"01:30.320 ","End":"01:35.210","Text":"put n=0 and cosine of 0 x is 1,"},{"Start":"01:35.210 ","End":"01:38.115","Text":"so we just have this."},{"Start":"01:38.115 ","End":"01:43.725","Text":"This integral is 1 so we just have 2 over Pi for a_naught."},{"Start":"01:43.725 ","End":"01:45.765","Text":"Now we have a_n and a_naught."},{"Start":"01:45.765 ","End":"01:47.940","Text":"Let\u0027s go on to compute b_n,"},{"Start":"01:47.940 ","End":"01:50.180","Text":"b_n is given by this formula."},{"Start":"01:50.180 ","End":"01:52.760","Text":"Sine of n x is odd."},{"Start":"01:52.760 ","End":"01:56.840","Text":"As before we\u0027re taking the integral just from minus 1-1 because"},{"Start":"01:56.840 ","End":"02:01.774","Text":"of f. This is odd and so this is going to equal 0."},{"Start":"02:01.774 ","End":"02:03.815","Text":"Now we have a_n in general,"},{"Start":"02:03.815 ","End":"02:07.280","Text":"specifically a_0 and we have the general b_n."},{"Start":"02:07.280 ","End":"02:09.785","Text":"If we plug it into this formula,"},{"Start":"02:09.785 ","End":"02:10.970","Text":"all the b_n\u0027s are 0,"},{"Start":"02:10.970 ","End":"02:12.710","Text":"so this drops off."},{"Start":"02:12.710 ","End":"02:15.530","Text":"What we get is a_naught is 2 over Pi,"},{"Start":"02:15.530 ","End":"02:17.630","Text":"so this is 1 over Pi."},{"Start":"02:17.630 ","End":"02:22.115","Text":"In general, we\u0027ve lost a_n but this is what it was."},{"Start":"02:22.115 ","End":"02:25.820","Text":"2 over Pi sine n over n,"},{"Start":"02:25.820 ","End":"02:27.860","Text":"2 over Pi sine over n,"},{"Start":"02:27.860 ","End":"02:29.875","Text":"and then cosine nx."},{"Start":"02:29.875 ","End":"02:34.020","Text":"This is the answer and we are done."}],"ID":28718}],"Thumbnail":null,"ID":294450},{"Name":"Complex Fourier Series","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Simple Introduction","Duration":"5m 57s","ChapterTopicVideoID":27522,"CourseChapterTopicPlaylistID":294451,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"Now we\u0027ve talked about Fourier Series of Real Functions."},{"Start":"00:04.140 ","End":"00:09.105","Text":"In this clip, we\u0027ll introduce Fourier Series for Complex Functions."},{"Start":"00:09.105 ","End":"00:10.695","Text":"Just to remind you,"},{"Start":"00:10.695 ","End":"00:14.010","Text":"this is the Fourier series for a real function."},{"Start":"00:14.010 ","End":"00:15.855","Text":"It has basically 2 parts,"},{"Start":"00:15.855 ","End":"00:19.740","Text":"sine and cosine, and also a constant bit."},{"Start":"00:19.740 ","End":"00:24.210","Text":"Although the constant 1 could be thought of as cosine 0x."},{"Start":"00:24.210 ","End":"00:29.430","Text":"The formulas to compute the coefficients an and the coefficients b_n."},{"Start":"00:29.430 ","End":"00:31.930","Text":"In fact, I\u0027ll remind you,"},{"Start":"00:32.360 ","End":"00:35.940","Text":"this is the formula that we have."},{"Start":"00:35.940 ","End":"00:39.465","Text":"In contrast for complex numbers,"},{"Start":"00:39.465 ","End":"00:42.785","Text":"and maybe I\u0027m spoiling the result I\u0027ll give it to you now,"},{"Start":"00:42.785 ","End":"00:46.340","Text":"the Fourier series expands like so,"},{"Start":"00:46.340 ","End":"00:49.060","Text":"just 1 set of coefficients c_n,"},{"Start":"00:49.060 ","End":"00:54.810","Text":"and the building blocks are e^ix instead of the cosine and the sine."},{"Start":"00:54.810 ","End":"00:56.835","Text":"Need to compute c_n."},{"Start":"00:56.835 ","End":"01:00.005","Text":"Let\u0027s work to get the formula for that."},{"Start":"01:00.005 ","End":"01:02.045","Text":"Recall Euler\u0027s formula,"},{"Start":"01:02.045 ","End":"01:05.865","Text":"and if we put Theta=nx,"},{"Start":"01:05.865 ","End":"01:09.845","Text":"then we get a formula for e^inx as follows."},{"Start":"01:09.845 ","End":"01:12.245","Text":"If we put Theta is minus nx,"},{"Start":"01:12.245 ","End":"01:14.060","Text":"then we get a similar formula,"},{"Start":"01:14.060 ","End":"01:17.870","Text":"e^minus inx and cosine is an even function,"},{"Start":"01:17.870 ","End":"01:18.930","Text":"sine is an odd function,"},{"Start":"01:18.930 ","End":"01:20.795","Text":"so the minus goes here."},{"Start":"01:20.795 ","End":"01:24.625","Text":"If we add these 2 equations and divide by 2,"},{"Start":"01:24.625 ","End":"01:32.250","Text":"then we get e^inx plus e^minus inx over 2 is just the cosine nx."},{"Start":"01:32.250 ","End":"01:37.130","Text":"If we subtract this minus this but divided by 2i,"},{"Start":"01:37.130 ","End":"01:40.685","Text":"then we get the other formula."},{"Start":"01:40.685 ","End":"01:44.840","Text":"Now we have cosine nx and sine nx in terms of the exponential."},{"Start":"01:44.840 ","End":"01:47.690","Text":"Now here\u0027s the real form of the Fourier series,"},{"Start":"01:47.690 ","End":"01:51.100","Text":"and we\u0027ll use this to derive the complex form."},{"Start":"01:51.100 ","End":"01:54.320","Text":"We\u0027ll just make some substitutions to substitute"},{"Start":"01:54.320 ","End":"01:57.995","Text":"cosine x and sine nx from these 2 formulas."},{"Start":"01:57.995 ","End":"02:04.445","Text":"So cosine nx, and sine nx is this."},{"Start":"02:04.445 ","End":"02:06.620","Text":"Then we just do some algebra."},{"Start":"02:06.620 ","End":"02:12.695","Text":"If we collect together the e^inx from here and here,"},{"Start":"02:12.695 ","End":"02:17.075","Text":"its coefficient will be a half and then there\u0027ll be an from here,"},{"Start":"02:17.075 ","End":"02:20.395","Text":"and b_n over 2i,"},{"Start":"02:20.395 ","End":"02:23.300","Text":"besides the half is another i in the denominator."},{"Start":"02:23.300 ","End":"02:27.335","Text":"And if we gather the e^minus inx from here,"},{"Start":"02:27.335 ","End":"02:30.050","Text":"and from here, let me get something similar."},{"Start":"02:30.050 ","End":"02:34.100","Text":"We get an minus 1 over ib_n for this."},{"Start":"02:34.100 ","End":"02:38.265","Text":"Yeah, just might as well do the highlighting. So you can see."},{"Start":"02:38.265 ","End":"02:42.990","Text":"The next step is just to replace 1 over i by minus i,"},{"Start":"02:42.990 ","End":"02:44.865","Text":"because i squared is minus 1,"},{"Start":"02:44.865 ","End":"02:47.155","Text":"and minus 1 over i is plus i."},{"Start":"02:47.155 ","End":"02:49.000","Text":"For the second sum,"},{"Start":"02:49.000 ","End":"02:53.980","Text":"what we can do is replace n by minus n. Want to get rid of this minus here,"},{"Start":"02:53.980 ","End":"02:55.930","Text":"make it more uniform."},{"Start":"02:55.930 ","End":"02:59.530","Text":"You have to replace n by minus n and the sum also,"},{"Start":"02:59.530 ","End":"03:03.130","Text":"so it will be from not minus 1 to minus infinity,"},{"Start":"03:03.130 ","End":"03:06.145","Text":"but the other way round, minus infinity to minus 1."},{"Start":"03:06.145 ","End":"03:12.515","Text":"Instead of n here and here we put minus n and the minus here becomes a plus."},{"Start":"03:12.515 ","End":"03:19.520","Text":"For consistency, the constant term we can write as e^i 0 x and is 0 here."},{"Start":"03:19.520 ","End":"03:21.995","Text":"Now I said we are going for this form."},{"Start":"03:21.995 ","End":"03:24.290","Text":"We just have to compare coefficients."},{"Start":"03:24.290 ","End":"03:28.970","Text":"We have 3 types of n for this sum n bigger equal to 1 here"},{"Start":"03:28.970 ","End":"03:35.010","Text":"n less than or equal to minus 1, and here n=0."},{"Start":"03:35.350 ","End":"03:41.705","Text":"For negative n, we get this formula here."},{"Start":"03:41.705 ","End":"03:44.525","Text":"Let\u0027s summarize what we have so far."},{"Start":"03:44.525 ","End":"03:49.790","Text":"We express f of x in the complex Fourier form,"},{"Start":"03:49.790 ","End":"03:51.250","Text":"as a sum of this series,"},{"Start":"03:51.250 ","End":"03:53.180","Text":"and we found that if this is so,"},{"Start":"03:53.180 ","End":"03:57.365","Text":"we have to have c_n equals this when n is bigger or equal to 1,"},{"Start":"03:57.365 ","End":"04:00.085","Text":"this when n is 0, this when n is negative."},{"Start":"04:00.085 ","End":"04:03.380","Text":"Let\u0027s try and combine these 3 forms into a single form."},{"Start":"04:03.380 ","End":"04:06.290","Text":"So let\u0027s take first of all N bigger or equal to 1."},{"Start":"04:06.290 ","End":"04:07.790","Text":"If that\u0027s the case,"},{"Start":"04:07.790 ","End":"04:10.550","Text":"then our formula is this."},{"Start":"04:10.550 ","End":"04:13.535","Text":"But we know what an and b_n are,"},{"Start":"04:13.535 ","End":"04:17.270","Text":"we had that just earlier that an from"},{"Start":"04:17.270 ","End":"04:21.695","Text":"the real case is equal to what I\u0027ve colored here in red and b_n,"},{"Start":"04:21.695 ","End":"04:24.075","Text":"what I\u0027ve colored here in blue."},{"Start":"04:24.075 ","End":"04:26.425","Text":"I\u0027ll do some algebra."},{"Start":"04:26.425 ","End":"04:29.655","Text":"We can write this as 1 over 2Pi,"},{"Start":"04:29.655 ","End":"04:34.265","Text":"and also we can combine with the distributive law and so on."},{"Start":"04:34.265 ","End":"04:38.870","Text":"F of x times cosine nx minus i sine nx,"},{"Start":"04:38.870 ","End":"04:43.880","Text":"and the 2 in both cases gets put together with the Pi. We have this."},{"Start":"04:43.880 ","End":"04:47.795","Text":"Now this means that c_n is equal to,"},{"Start":"04:47.795 ","End":"04:51.380","Text":"instead of cosine nx minus i sine nx,"},{"Start":"04:51.380 ","End":"04:54.920","Text":"this is equal to e^minus inx."},{"Start":"04:54.920 ","End":"04:58.580","Text":"This is what c_n is equal to when n is bigger or equal to 1."},{"Start":"04:58.580 ","End":"05:02.795","Text":"Now let\u0027s take the case where n is less than or equal to minus 1."},{"Start":"05:02.795 ","End":"05:04.540","Text":"This is the formula we had."},{"Start":"05:04.540 ","End":"05:07.190","Text":"Then again substituting, I\u0027ll go more quickly this time."},{"Start":"05:07.190 ","End":"05:08.675","Text":"Similar idea."},{"Start":"05:08.675 ","End":"05:13.985","Text":"Once again, we have that this is e^minus inx."},{"Start":"05:13.985 ","End":"05:16.895","Text":"Exactly the same formula here and here."},{"Start":"05:16.895 ","End":"05:19.775","Text":"Let\u0027s see what happens if n is 0."},{"Start":"05:19.775 ","End":"05:23.750","Text":"In this case, we get the formula for a naught,"},{"Start":"05:23.750 ","End":"05:26.750","Text":"which is this, and it comes out to be this."},{"Start":"05:26.750 ","End":"05:31.109","Text":"If we throw in 1e^minus i0x,"},{"Start":"05:31.109 ","End":"05:36.200","Text":"this looks very much like this with n=0."},{"Start":"05:36.200 ","End":"05:39.080","Text":"So also c_n is equal to this."},{"Start":"05:39.080 ","End":"05:40.400","Text":"If you just let n=0,"},{"Start":"05:40.400 ","End":"05:43.370","Text":"then we have all 3 cases."},{"Start":"05:43.370 ","End":"05:45.090","Text":"We have the same formula."},{"Start":"05:45.090 ","End":"05:50.925","Text":"That means that we can write our c_n as follows,"},{"Start":"05:50.925 ","End":"05:53.220","Text":"and that\u0027s what we were looking for,"},{"Start":"05:53.220 ","End":"05:57.430","Text":"and that completes this clip."}],"ID":28719},{"Watched":false,"Name":"Example","Duration":"2m 8s","ChapterTopicVideoID":27523,"CourseChapterTopicPlaylistID":294451,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.515","Text":"Now an example how to compute the Fourier series of a complex function."},{"Start":"00:05.515 ","End":"00:10.965","Text":"In fact, we\u0027ll take the same example that we worked earlier with real Fourier series,"},{"Start":"00:10.965 ","End":"00:15.660","Text":"f(x)=x on the interval from minus Pi to Pi."},{"Start":"00:15.660 ","End":"00:17.765","Text":"This is what it looks like,"},{"Start":"00:17.765 ","End":"00:18.920","Text":"just the middle bit."},{"Start":"00:18.920 ","End":"00:22.775","Text":"Unless we repeated periodically with period of 2Pi,"},{"Start":"00:22.775 ","End":"00:25.690","Text":"this is the f(x)=x."},{"Start":"00:25.690 ","End":"00:31.340","Text":"Here\u0027s a reminder of the complex Fourier series expansion."},{"Start":"00:31.340 ","End":"00:37.962","Text":"f(x) is represented by this series where c_n is given by this integral formula,"},{"Start":"00:37.962 ","End":"00:40.235","Text":"it\u0027s also an inner product anyway."},{"Start":"00:40.235 ","End":"00:44.534","Text":"This is the formula and we just have to compute it in our case."},{"Start":"00:44.534 ","End":"00:50.285","Text":"Now, we distinguish n=0 and then not equals 0, you\u0027ll see why."},{"Start":"00:50.285 ","End":"00:55.640","Text":"Just copying this with f(x)=x, we get this."},{"Start":"00:55.640 ","End":"00:58.100","Text":"Now, use integration by parts."},{"Start":"00:58.100 ","End":"01:00.050","Text":"This is the formula."},{"Start":"01:00.050 ","End":"01:02.300","Text":"We\u0027ll let x be v,"},{"Start":"01:02.300 ","End":"01:06.185","Text":"and e^minus inx will be u\u0027."},{"Start":"01:06.185 ","End":"01:10.805","Text":"That means that u is this here,"},{"Start":"01:10.805 ","End":"01:13.040","Text":"but it wouldn\u0027t make sense if n was 0."},{"Start":"01:13.040 ","End":"01:16.010","Text":"That\u0027s why we included n=0."},{"Start":"01:16.010 ","End":"01:20.510","Text":"Anyway, let\u0027s not get bogged down in the technical details,"},{"Start":"01:20.510 ","End":"01:23.454","Text":"just show you the development."},{"Start":"01:23.454 ","End":"01:33.380","Text":"Eventually, we get that c_n is equal to i times minus 1^n over n, and n is not 0."},{"Start":"01:33.380 ","End":"01:39.940","Text":"What is c_0? c_0 is the integral of xdx."},{"Start":"01:39.940 ","End":"01:42.140","Text":"x is an odd function."},{"Start":"01:42.140 ","End":"01:44.210","Text":"This is a symmetric interval,"},{"Start":"01:44.210 ","End":"01:47.530","Text":"so this is 0. We don\u0027t need c_0."},{"Start":"01:47.530 ","End":"01:50.985","Text":"So x is the following series;"},{"Start":"01:50.985 ","End":"01:53.520","Text":"c_n is this,"},{"Start":"01:53.520 ","End":"01:57.445","Text":"e^inx, sum from minus infinity to infinity."},{"Start":"01:57.445 ","End":"01:58.770","Text":"I don\u0027t know what the notation is,"},{"Start":"01:58.770 ","End":"02:00.700","Text":"but I\u0027ll just write it, n is not 0."},{"Start":"02:00.700 ","End":"02:03.800","Text":"It\u0027s the sum except for n=0,"},{"Start":"02:03.800 ","End":"02:05.450","Text":"and it wouldn\u0027t make sense anyway."},{"Start":"02:05.450 ","End":"02:08.850","Text":"That concludes this example exercise."}],"ID":28720},{"Watched":false,"Name":"Advanced Introduction","Duration":"1m 40s","ChapterTopicVideoID":27521,"CourseChapterTopicPlaylistID":294451,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"This is the second part of the introduction."},{"Start":"00:02.970 ","End":"00:07.810","Text":"Although really it\u0027s a summary rather than an introduction."},{"Start":"00:07.940 ","End":"00:10.965","Text":"We have an inner product space,"},{"Start":"00:10.965 ","End":"00:13.200","Text":"which is a vector space."},{"Start":"00:13.200 ","End":"00:20.250","Text":"It\u0027s the space of all piecewise continuous functions on the interval minus Pi to Pi."},{"Start":"00:20.250 ","End":"00:23.985","Text":"I suppose I should have said complex functions."},{"Start":"00:23.985 ","End":"00:26.670","Text":"Although it should be clear from the context."},{"Start":"00:26.670 ","End":"00:29.175","Text":"Just like in the real case,"},{"Start":"00:29.175 ","End":"00:34.800","Text":"we can call it E instead of L^2 PC, it\u0027s shorter."},{"Start":"00:34.800 ","End":"00:37.500","Text":"Now it has an inner product,"},{"Start":"00:37.500 ","End":"00:39.900","Text":"which is the following."},{"Start":"00:39.900 ","End":"00:42.365","Text":"As opposed to the real case,"},{"Start":"00:42.365 ","End":"00:45.530","Text":"it\u0027s important here to have the conjugate,"},{"Start":"00:45.530 ","End":"00:48.460","Text":"this bar over the second function."},{"Start":"00:48.460 ","End":"00:52.270","Text":"Now it has an infinite orthonormal system."},{"Start":"00:52.270 ","End":"00:55.610","Text":"It\u0027s more symmetric, it\u0027s cleaner than in the real case,"},{"Start":"00:55.610 ","End":"00:57.680","Text":"there we had some constant,"},{"Start":"00:57.680 ","End":"00:59.420","Text":"and sines, and cosines,"},{"Start":"00:59.420 ","End":"01:01.895","Text":"here we just have e^inx,"},{"Start":"01:01.895 ","End":"01:04.250","Text":"but the series is doubly infinity."},{"Start":"01:04.250 ","End":"01:06.980","Text":"It goes from minus infinity to infinity."},{"Start":"01:06.980 ","End":"01:13.910","Text":"The complex Fourier series of a function f is given by the double sum or"},{"Start":"01:13.910 ","End":"01:21.395","Text":"doubly infinite sum from minus infinity to plus infinity of c_n e^inx."},{"Start":"01:21.395 ","End":"01:25.410","Text":"There\u0027s a formula for the coefficient c_n and it\u0027s"},{"Start":"01:25.410 ","End":"01:29.615","Text":"given by c_n is the inner product of f with e^inx."},{"Start":"01:29.615 ","End":"01:32.960","Text":"That\u0027s an integral. This is what it\u0027s equal to."},{"Start":"01:32.960 ","End":"01:36.775","Text":"N goes from minus infinity to infinity."},{"Start":"01:36.775 ","End":"01:41.110","Text":"That\u0027s all I want to say for this summary. We\u0027re done."}],"ID":28721},{"Watched":false,"Name":"Exercise 1","Duration":"3m 37s","ChapterTopicVideoID":27524,"CourseChapterTopicPlaylistID":294451,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"In this exercise,"},{"Start":"00:02.040 ","End":"00:05.550","Text":"we\u0027re going to compute the complex Fourier series for"},{"Start":"00:05.550 ","End":"00:10.245","Text":"the function f(x) on the interval from minus Pi to Pi."},{"Start":"00:10.245 ","End":"00:13.170","Text":"This is what f(x) is equal to."},{"Start":"00:13.170 ","End":"00:17.625","Text":"It\u0027s defined piecewise and there\u0027s a picture if it helps."},{"Start":"00:17.625 ","End":"00:23.550","Text":"The general form f(x) is represented by the series,"},{"Start":"00:23.550 ","End":"00:27.960","Text":"the sum c_n e^inx and the n goes from minus infinity to"},{"Start":"00:27.960 ","End":"00:33.420","Text":"infinity and the formula for c_n is this integral."},{"Start":"00:33.420 ","End":"00:42.600","Text":"Let\u0027s compute it for each n. This is equal to f(x) is minus x from minus Pi to 0,"},{"Start":"00:42.600 ","End":"00:44.540","Text":"and 0 outside of that,"},{"Start":"00:44.540 ","End":"00:50.480","Text":"so we just need the integral from minus Pi to 0 of minus x, e^minus inx."},{"Start":"00:50.480 ","End":"00:52.670","Text":"We\u0027ll do this by parts,"},{"Start":"00:52.670 ","End":"00:54.290","Text":"put the minus in front,"},{"Start":"00:54.290 ","End":"00:57.350","Text":"then we have x times e^minus inx, this will be u,"},{"Start":"00:57.350 ","End":"01:06.380","Text":"this will be v\u0027 and u\u0027 will be 1 and v will be e^minus inx over minus in."},{"Start":"01:06.380 ","End":"01:08.855","Text":"This is u times v,"},{"Start":"01:08.855 ","End":"01:13.190","Text":"and this is the integral of vdu."},{"Start":"01:13.190 ","End":"01:18.170","Text":"Note that this won\u0027t work for n=0 because we have an n in the denominator."},{"Start":"01:18.170 ","End":"01:21.950","Text":"It\u0027s true for c_n when n is not equal to 0,"},{"Start":"01:21.950 ","End":"01:25.010","Text":"and we\u0027ll compute c_0 later."},{"Start":"01:25.010 ","End":"01:28.775","Text":"This is equal to integral of this,"},{"Start":"01:28.775 ","End":"01:31.070","Text":"again, we divide by minus in,"},{"Start":"01:31.070 ","End":"01:33.545","Text":"so we get minus in^2,"},{"Start":"01:33.545 ","End":"01:38.470","Text":"and minus (in)^2 is i^2 n^2,"},{"Start":"01:38.470 ","End":"01:40.530","Text":"i^2 is minus 1."},{"Start":"01:40.530 ","End":"01:44.720","Text":"That changes that minus into a plus and we just have n^2."},{"Start":"01:44.720 ","End":"01:46.655","Text":"Here, for convenience,"},{"Start":"01:46.655 ","End":"01:50.900","Text":"change the limits of 0 and minus Pi,"},{"Start":"01:50.900 ","End":"01:54.095","Text":"then we can get rid of that minus here."},{"Start":"01:54.095 ","End":"01:59.210","Text":"What we get is you plug in 0, it\u0027s 0,"},{"Start":"01:59.210 ","End":"02:00.845","Text":"plug in minus Pi,"},{"Start":"02:00.845 ","End":"02:06.610","Text":"we have minus Pi e^plus in Pi over in."},{"Start":"02:06.610 ","End":"02:10.245","Text":"For this, when x is 0,"},{"Start":"02:10.245 ","End":"02:11.655","Text":"we get 1,"},{"Start":"02:11.655 ","End":"02:14.280","Text":"and when x is minus Pi,"},{"Start":"02:14.280 ","End":"02:17.575","Text":"we have e^in Pi,"},{"Start":"02:17.575 ","End":"02:21.605","Text":"which we know is minus 1^ n,"},{"Start":"02:21.605 ","End":"02:22.745","Text":"and the minus here,"},{"Start":"02:22.745 ","End":"02:25.535","Text":"because we subtract this minus this."},{"Start":"02:25.535 ","End":"02:31.045","Text":"Now let\u0027s compute c_0 because this wasn\u0027t true for n=0."},{"Start":"02:31.045 ","End":"02:35.010","Text":"Copying the formula for c_n from here,"},{"Start":"02:35.010 ","End":"02:37.320","Text":"if we had n=0,"},{"Start":"02:37.320 ","End":"02:40.815","Text":"then e^minus inx is e^0,"},{"Start":"02:40.815 ","End":"02:41.990","Text":"which is just 1."},{"Start":"02:41.990 ","End":"02:44.390","Text":"So we\u0027re left with x dx."},{"Start":"02:44.390 ","End":"02:48.905","Text":"This is equal to integral of x is x^2 over 2,"},{"Start":"02:48.905 ","End":"02:53.930","Text":"and we invert the order of the substitution limits,"},{"Start":"02:53.930 ","End":"02:56.485","Text":"and then we can get rid of the minus here."},{"Start":"02:56.485 ","End":"03:01.425","Text":"We get minus Pi^2 over 2 minus 0."},{"Start":"03:01.425 ","End":"03:04.410","Text":"This just comes out to be Pi over 4."},{"Start":"03:04.410 ","End":"03:08.630","Text":"Now we have c_n for all n. Now we just have to plug it"},{"Start":"03:08.630 ","End":"03:13.220","Text":"in to the formula for the series, which is this."},{"Start":"03:13.220 ","End":"03:18.560","Text":"In our case it comes out to be c naught times e^i0 Pi,"},{"Start":"03:18.560 ","End":"03:20.555","Text":"but that\u0027s just 1,"},{"Start":"03:20.555 ","End":"03:30.160","Text":"plus the sum from minus infinity to infinity excluding 0 of what we had here, e^inx."},{"Start":"03:30.160 ","End":"03:32.510","Text":"Possibly this could be simplified,"},{"Start":"03:32.510 ","End":"03:34.205","Text":"but we needn\u0027t bother,"},{"Start":"03:34.205 ","End":"03:37.590","Text":"and this is our answer and we\u0027re done."}],"ID":28722},{"Watched":false,"Name":"Exercise 2","Duration":"3m 18s","ChapterTopicVideoID":27525,"CourseChapterTopicPlaylistID":294451,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this exercise, we\u0027re going to compute the complex Fourier series for"},{"Start":"00:04.080 ","End":"00:08.115","Text":"the function f(x) on the interval from minus Pi to Pi."},{"Start":"00:08.115 ","End":"00:14.235","Text":"The function f is given piecewise as x in this part and 2x in this part."},{"Start":"00:14.235 ","End":"00:17.895","Text":"Here it\u0027s x and here it\u0027s 2x."},{"Start":"00:17.895 ","End":"00:21.060","Text":"The formula for the coefficient C_n,"},{"Start":"00:21.060 ","End":"00:28.065","Text":"in the complex expansion is 1 over 2 Pi the integral f(x) e^minus inx dx."},{"Start":"00:28.065 ","End":"00:30.120","Text":"Now f(x) is defined piecewise,"},{"Start":"00:30.120 ","End":"00:32.160","Text":"so we break it up into 2 integrals."},{"Start":"00:32.160 ","End":"00:39.720","Text":"It will be x e^minus inx from minus Pi to 0 and 2x here."},{"Start":"00:39.720 ","End":"00:42.150","Text":"Now you don\u0027t see the 2x because the 2 here,"},{"Start":"00:42.150 ","End":"00:44.660","Text":"cancels out with the 2 here."},{"Start":"00:44.660 ","End":"00:48.575","Text":"This is what we get. Now it\u0027s the same integrand in both."},{"Start":"00:48.575 ","End":"00:50.390","Text":"If we just compute 1 indefinite"},{"Start":"00:50.390 ","End":"00:54.200","Text":"integral and we can use it and substitute different limits."},{"Start":"00:54.200 ","End":"01:01.150","Text":"Now we did it in a previous exercise and the integral of this comes out to be this."},{"Start":"01:01.150 ","End":"01:06.395","Text":"If we take e^minus inx out the brackets and we\u0027re left with this."},{"Start":"01:06.395 ","End":"01:12.070","Text":"After we change the order and pull the minus out, we get this."},{"Start":"01:12.070 ","End":"01:15.530","Text":"We just stick this in here."},{"Start":"01:15.530 ","End":"01:17.390","Text":"Once from minus Pi to 0,"},{"Start":"01:17.390 ","End":"01:22.650","Text":"once from 0 to Pi and here it\u0027s 1 over 2 Pi and here it\u0027s 1 over Pi."},{"Start":"01:22.650 ","End":"01:26.375","Text":"Now here when we plug in 0, this is 0,"},{"Start":"01:26.375 ","End":"01:32.110","Text":"this is 1, so we get 1 over n squared and similarly we have minus 1 over n squared here."},{"Start":"01:32.110 ","End":"01:34.720","Text":"When x is minus Pi,"},{"Start":"01:34.720 ","End":"01:38.180","Text":"we just copy this with a minus Pi here."},{"Start":"01:38.180 ","End":"01:40.940","Text":"Here it\u0027s going to be plus Pi."},{"Start":"01:40.940 ","End":"01:44.510","Text":"Similarly for this term on it\u0027s going to be"},{"Start":"01:44.510 ","End":"01:51.220","Text":"a minus inPi and it\u0027s going to be a minus here without the minus here."},{"Start":"01:51.220 ","End":"01:54.110","Text":"This is what we get. An estimate is"},{"Start":"01:54.110 ","End":"01:58.280","Text":"just straightforward algebra, brackets, simplification."},{"Start":"01:58.280 ","End":"02:01.175","Text":"I won\u0027t go through it line by line."},{"Start":"02:01.175 ","End":"02:04.640","Text":"We get to this line and we\u0027ll stop here."},{"Start":"02:04.640 ","End":"02:06.620","Text":"Possibly it could be simplified further,"},{"Start":"02:06.620 ","End":"02:07.960","Text":"but let\u0027s leave it at this."},{"Start":"02:07.960 ","End":"02:09.350","Text":"This is c_n."},{"Start":"02:09.350 ","End":"02:12.470","Text":"Now, what we did here I should have mentioned,"},{"Start":"02:12.470 ","End":"02:18.110","Text":"is not true when n is 0 because we can\u0027t divide by 0 so we have"},{"Start":"02:18.110 ","End":"02:24.385","Text":"a formula only when n is not 0 and we need to compute c_naught separately."},{"Start":"02:24.385 ","End":"02:28.115","Text":"Going back and plugging in n=0,"},{"Start":"02:28.115 ","End":"02:31.790","Text":"e^minus inx is 1,"},{"Start":"02:31.790 ","End":"02:33.515","Text":"and this is what we get."},{"Start":"02:33.515 ","End":"02:37.730","Text":"The integral of x in each case is x squared over 2."},{"Start":"02:37.730 ","End":"02:42.755","Text":"Just algebra comes out to be Pi over 4."},{"Start":"02:42.755 ","End":"02:45.200","Text":"When n is not 0, we get this."},{"Start":"02:45.200 ","End":"02:47.465","Text":"When n is 0, we have this."},{"Start":"02:47.465 ","End":"02:51.650","Text":"Now we just have to substitute in this formula for the representation as"},{"Start":"02:51.650 ","End":"03:01.095","Text":"a complex Fourier series is just the Pi over 4 here for c_naught each of the i0x,"},{"Start":"03:01.095 ","End":"03:05.180","Text":"which is 1 and then the sum from minus infinity to"},{"Start":"03:05.180 ","End":"03:09.335","Text":"infinity excluding 0 of whatever it was that we had here,"},{"Start":"03:09.335 ","End":"03:11.440","Text":"just plug it down here."},{"Start":"03:11.440 ","End":"03:14.140","Text":"That\u0027s e^inx."},{"Start":"03:14.140 ","End":"03:18.010","Text":"That concludes this exercise."}],"ID":28723},{"Watched":false,"Name":"Exercise 3","Duration":"3m 5s","ChapterTopicVideoID":27526,"CourseChapterTopicPlaylistID":294451,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.320 ","End":"00:03.840","Text":"Here we\u0027re going to compute the complex Fourier series for"},{"Start":"00:03.840 ","End":"00:07.350","Text":"the function f(x) on the interval from minus Pi to Pi,"},{"Start":"00:07.350 ","End":"00:09.375","Text":"where f is given as follows."},{"Start":"00:09.375 ","End":"00:16.035","Text":"It\u0027s equal to 1 between minus Pi and 0 and equal minus 2 between 0 and Pi."},{"Start":"00:16.035 ","End":"00:18.762","Text":"Straightaway, start computing the coefficient,"},{"Start":"00:18.762 ","End":"00:21.915","Text":"c_n, using this formula."},{"Start":"00:21.915 ","End":"00:25.320","Text":"We break it up into 2 integrals because it\u0027s piecewise."},{"Start":"00:25.320 ","End":"00:27.405","Text":"We have from minus Pi to 0,"},{"Start":"00:27.405 ","End":"00:31.215","Text":"1 and from 0 to Pi minus 2."},{"Start":"00:31.215 ","End":"00:34.620","Text":"This is equal to throughout the 1,"},{"Start":"00:34.620 ","End":"00:37.740","Text":"pull the minus 2 in front of the integral."},{"Start":"00:37.740 ","End":"00:39.000","Text":"So the minus is here,"},{"Start":"00:39.000 ","End":"00:40.695","Text":"and the 2 cancels with the 2."},{"Start":"00:40.695 ","End":"00:43.805","Text":"This is what we have and this is equal to,"},{"Start":"00:43.805 ","End":"00:45.710","Text":"this is the same integrand in both cases."},{"Start":"00:45.710 ","End":"00:48.650","Text":"In each case we have each of the minus inx over minus"},{"Start":"00:48.650 ","End":"00:52.675","Text":"in but once between these limits and once between these limits."},{"Start":"00:52.675 ","End":"00:55.005","Text":"Of course n is not 0."},{"Start":"00:55.005 ","End":"00:58.160","Text":"We\u0027ll have to compute c_0 separately later."},{"Start":"00:58.160 ","End":"01:02.570","Text":"Anyway, this is equal to switch the limits around to get rid of"},{"Start":"01:02.570 ","End":"01:07.820","Text":"this minus and also combine this minus with this minus, everything\u0027s plus."},{"Start":"01:07.820 ","End":"01:10.355","Text":"Substitute the limits."},{"Start":"01:10.355 ","End":"01:13.230","Text":"We have 1 over 2 Pi times this,"},{"Start":"01:13.230 ","End":"01:15.530","Text":"plus 1 over Pi times this."},{"Start":"01:15.530 ","End":"01:18.050","Text":"Well, it\u0027s not the same, but they are the same really because each of"},{"Start":"01:18.050 ","End":"01:20.810","Text":"the in Pi is minus 1^n,"},{"Start":"01:20.810 ","End":"01:22.990","Text":"and this is minus 1^n,"},{"Start":"01:22.990 ","End":"01:25.470","Text":"or 1 over that same thing."},{"Start":"01:25.470 ","End":"01:29.295","Text":"It\u0027s minus 1^n minus 1 over in here and here."},{"Start":"01:29.295 ","End":"01:34.095","Text":"We collect them, 1 over 2 Pi plus 1 over Pi is 3 over 2 Pi."},{"Start":"01:34.095 ","End":"01:36.780","Text":"We have 3 over 2 Pi times this."},{"Start":"01:36.780 ","End":"01:40.999","Text":"That\u0027s our c_n for when n is not equal to 0."},{"Start":"01:40.999 ","End":"01:46.905","Text":"Now, going back to c_n from back here,"},{"Start":"01:46.905 ","End":"01:50.385","Text":"just plug in n=0."},{"Start":"01:50.385 ","End":"01:52.920","Text":"When n is 0, this is 1 and this is 1."},{"Start":"01:52.920 ","End":"01:56.085","Text":"So this integral is Pi and this integral is Pi."},{"Start":"01:56.085 ","End":"01:58.740","Text":"We get Pi over 2 Pi minus Pi over Pi,"},{"Start":"01:58.740 ","End":"02:01.305","Text":"a half minus 1 minus a half."},{"Start":"02:01.305 ","End":"02:04.440","Text":"We have c_0 is this,"},{"Start":"02:04.440 ","End":"02:08.660","Text":"and c_n is this."},{"Start":"02:08.660 ","End":"02:11.810","Text":"Now we just have to plug them into the formula."},{"Start":"02:11.810 ","End":"02:13.115","Text":"This is what we get,"},{"Start":"02:13.115 ","End":"02:18.115","Text":"the minus a half from here and this coefficient from here."},{"Start":"02:18.115 ","End":"02:20.615","Text":"This is true for n not equal to 0."},{"Start":"02:20.615 ","End":"02:25.790","Text":"But notice that if n is even here,"},{"Start":"02:25.790 ","End":"02:27.720","Text":"then minus 1^n is 1."},{"Start":"02:27.720 ","End":"02:29.610","Text":"This comes out to be 0."},{"Start":"02:29.610 ","End":"02:32.225","Text":"We only need the odd n here."},{"Start":"02:32.225 ","End":"02:36.410","Text":"Altogether, n is 0 or n is odd i.e.,"},{"Start":"02:36.410 ","End":"02:40.010","Text":"2k minus 1 for some k. What we can"},{"Start":"02:40.010 ","End":"02:43.895","Text":"say is that the Fourier series is the same minus a half,"},{"Start":"02:43.895 ","End":"02:47.630","Text":"but here replace n by 2k minus 1,"},{"Start":"02:47.630 ","End":"02:49.640","Text":"and k could be anything."},{"Start":"02:49.640 ","End":"02:52.250","Text":"I mean, 2k minus 1 can\u0027t be 0."},{"Start":"02:52.250 ","End":"02:56.550","Text":"We have this, but instead of n, 2k minus 1."},{"Start":"02:56.550 ","End":"02:58.860","Text":"Here, it\u0027s going to be minus 1,"},{"Start":"02:58.860 ","End":"03:00.345","Text":"minus 1, which is minus 2."},{"Start":"03:00.345 ","End":"03:02.865","Text":"Again here n is 2k minus 1."},{"Start":"03:02.865 ","End":"03:06.280","Text":"This is the answer. We are done."}],"ID":28724}],"Thumbnail":null,"ID":294451},{"Name":"Parseval Identity","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Real Parseval Identity - Introduction","Duration":"3m 41s","ChapterTopicVideoID":27537,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/27537.jpeg","UploadDate":"2021-12-01T10:54:43.3770000","DurationForVideoObject":"PT3M41S","Description":null,"MetaTitle":"Real Parseval Identity - Introduction: Video + Workbook | Proprep","MetaDescription":"Fourier Series - Parseval Identity. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/fourier-series/parseval-identity/vid28725","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"In this clip and the following one,"},{"Start":"00:02.460 ","End":"00:04.710","Text":"we\u0027ll talk about Parseval\u0027s identity,"},{"Start":"00:04.710 ","End":"00:06.435","Text":"parts of those theorem."},{"Start":"00:06.435 ","End":"00:10.620","Text":"In this clip, we\u0027ll talk about the case for real functions and the following one,"},{"Start":"00:10.620 ","End":"00:12.630","Text":"we\u0027ll talk about complex functions."},{"Start":"00:12.630 ","End":"00:16.275","Text":"Let\u0027s start with the theorem which is an identity."},{"Start":"00:16.275 ","End":"00:21.270","Text":"Suppose that f is piecewise continuous on minus Pi Pi,"},{"Start":"00:21.270 ","End":"00:24.855","Text":"and also it belongs to L^2 integrable"},{"Start":"00:24.855 ","End":"00:30.555","Text":"and it has a Fourier series as follows, familiar form."},{"Start":"00:30.555 ","End":"00:35.640","Text":"Then we have Parseval\u0027s identity, which is this."},{"Start":"00:35.640 ","End":"00:38.145","Text":"We won\u0027t read it out, it\u0027s just as written."},{"Start":"00:38.145 ","End":"00:41.525","Text":"Note that a_0 is an exceptional term."},{"Start":"00:41.525 ","End":"00:43.100","Text":"Not only the odd one out here,"},{"Start":"00:43.100 ","End":"00:46.370","Text":"but you might think because everything here is squared,"},{"Start":"00:46.370 ","End":"00:48.560","Text":"that it would be a_0^2/4,"},{"Start":"00:48.560 ","End":"00:51.160","Text":"but no, the 2 is correct, it\u0027s not a 4."},{"Start":"00:51.160 ","End":"00:55.430","Text":"Parseval\u0027s theorem is useful both in theory and in practice."},{"Start":"00:55.430 ","End":"00:57.830","Text":"In theory, it could be helpful to prove other theorems."},{"Start":"00:57.830 ","End":"01:03.365","Text":"In practice, one of the main uses is in computing certain infinite series,"},{"Start":"01:03.365 ","End":"01:05.555","Text":"and we\u0027ll give an example of that."},{"Start":"01:05.555 ","End":"01:12.610","Text":"Example exercise, given that x is Fourier series is the following sum,"},{"Start":"01:12.610 ","End":"01:18.200","Text":"this is the real Fourier series on the interval from minus Pi to Pi,"},{"Start":"01:18.200 ","End":"01:24.040","Text":"we have to compute the sum from 1 to Infinity, 1^2."},{"Start":"01:24.040 ","End":"01:28.210","Text":"We know that this is a convergent series in Calculus 1,"},{"Start":"01:28.210 ","End":"01:29.780","Text":"you probably encountered it."},{"Start":"01:29.780 ","End":"01:32.180","Text":"It\u0027s what\u0027s called a P-series,"},{"Start":"01:32.180 ","End":"01:37.655","Text":"1 to the power of p. The sum converges when p is bigger than 1,"},{"Start":"01:37.655 ","End":"01:40.205","Text":"and 2 is bigger than 1, so we\u0027re okay."},{"Start":"01:40.205 ","End":"01:44.185","Text":"But we never actually talked about what the sum is."},{"Start":"01:44.185 ","End":"01:47.610","Text":"x has a Fourier series as follows,"},{"Start":"01:47.610 ","End":"01:49.275","Text":"that\u0027s the general form."},{"Start":"01:49.275 ","End":"01:51.965","Text":"In our case, if we look at it here,"},{"Start":"01:51.965 ","End":"02:01.740","Text":"we see that the b_n is equal to the coefficient of the sine and there is nothing else."},{"Start":"02:01.740 ","End":"02:05.235","Text":"The a_0 is 0 and the a_n is 0."},{"Start":"02:05.235 ","End":"02:07.800","Text":"Generally, all the a_n\u0027s are 0."},{"Start":"02:07.800 ","End":"02:10.290","Text":"Now this is Parseval\u0027s identity,"},{"Start":"02:10.290 ","End":"02:12.510","Text":"go back and look, this is what it is."},{"Start":"02:12.510 ","End":"02:15.290","Text":"If we plug in, substitute our numbers,"},{"Start":"02:15.290 ","End":"02:20.825","Text":"what we get f(x) is the function x and,"},{"Start":"02:20.825 ","End":"02:24.470","Text":"we get this integral and then a_0, a_n,0."},{"Start":"02:24.470 ","End":"02:26.860","Text":"We just have this sum."},{"Start":"02:26.860 ","End":"02:30.465","Text":"Absolute value of x^2 is the same as x^2."},{"Start":"02:30.465 ","End":"02:32.220","Text":"Because it\u0027s an even function,"},{"Start":"02:32.220 ","End":"02:35.465","Text":"we can simplify, just take it from 0 to Pi."},{"Start":"02:35.465 ","End":"02:39.780","Text":"It\u0027s on a symmetric interval and then we put a 2 here."},{"Start":"02:40.180 ","End":"02:43.640","Text":"What it\u0027s equal to a minus 1 to"},{"Start":"02:43.640 ","End":"02:46.370","Text":"the power of something doesn\u0027t make any difference in absolute value."},{"Start":"02:46.370 ","End":"02:49.855","Text":"We get 2^2 is 4^2."},{"Start":"02:49.855 ","End":"02:54.675","Text":"Then the integral of x^2 is x^3/3."},{"Start":"02:54.675 ","End":"02:58.115","Text":"We can pull the 4 out here in front of the series."},{"Start":"02:58.115 ","End":"03:02.250","Text":"Then this is equal to Pi^3/3."},{"Start":"03:02.250 ","End":"03:06.150","Text":"Let\u0027s switch sides, here\u0027s the Pi^3/3 here\u0027s the 2/Pi."},{"Start":"03:06.150 ","End":"03:09.185","Text":"The 4 we\u0027ll bring to the other side, it\u0027s a quarter."},{"Start":"03:09.185 ","End":"03:12.765","Text":"All we have left, we\u0027ve switched sides like I said,"},{"Start":"03:12.765 ","End":"03:16.050","Text":"is the sum of 1^2."},{"Start":"03:16.050 ","End":"03:19.320","Text":"Now 2/4 times 3 is 1/6."},{"Start":"03:19.320 ","End":"03:21.615","Text":"What we get is Pi^2/6."},{"Start":"03:21.615 ","End":"03:24.435","Text":"Of course your have Pi^3/Pi is Pi^2."},{"Start":"03:24.435 ","End":"03:26.325","Text":"What\u0027s that roughly?"},{"Start":"03:26.325 ","End":"03:30.225","Text":"Pi^2 is about 10, it\u0027s about 10/6,1.6,1.7."},{"Start":"03:30.225 ","End":"03:37.110","Text":"That\u0027s the sum of the series from 1 to infinity,"},{"Start":"03:37.110 ","End":"03:41.440","Text":"in case you ever wondered. Okay, we\u0027re done."}],"ID":28725},{"Watched":false,"Name":"Exercise 1","Duration":"2m 1s","ChapterTopicVideoID":27538,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.230","Text":"In this exercise, we\u0027re given that f(x) is the function 1,"},{"Start":"00:04.230 ","End":"00:09.615","Text":"0 like in the picture and we\u0027re given the Fourier expansion,"},{"Start":"00:09.615 ","End":"00:12.465","Text":"I believe we did it in the previous exercise."},{"Start":"00:12.465 ","End":"00:14.460","Text":"Given the Fourier expansion,"},{"Start":"00:14.460 ","End":"00:20.370","Text":"we have to prove that the following numerical series converges 2Pi squared over 8."},{"Start":"00:20.370 ","End":"00:21.690","Text":"This series, by the way,"},{"Start":"00:21.690 ","End":"00:26.700","Text":"if you write it out as 1 over 1^2 plus 1 over 3^2 plus 1 over 5^2 plus 1 over 7^2,"},{"Start":"00:26.700 ","End":"00:29.310","Text":"and so on. Let\u0027s solve it."},{"Start":"00:29.310 ","End":"00:33.044","Text":"Recall Parseval\u0027s identity that"},{"Start":"00:33.044 ","End":"00:36.630","Text":"we have this that the 1 over Pi integral of the absolute value"},{"Start":"00:36.630 ","End":"00:39.110","Text":"of the function squared is the sum of the squares of"},{"Start":"00:39.110 ","End":"00:43.760","Text":"all the coefficients with the exception of the a_0,"},{"Start":"00:43.760 ","End":"00:46.310","Text":"where we have to divide this by 2."},{"Start":"00:46.310 ","End":"00:50.890","Text":"We\u0027ll apply this identity to the Fourier series we were given."},{"Start":"00:50.890 ","End":"00:56.205","Text":"Notice that a_0 is 1 because this is a_0 over 2"},{"Start":"00:56.205 ","End":"01:02.460","Text":"and these are all b_ n or rather b 2k minus 1."},{"Start":"01:02.460 ","End":"01:04.820","Text":"Anyway, we take the sum of all the squares,"},{"Start":"01:04.820 ","End":"01:08.705","Text":"just being careful with the first one because it\u0027s slightly different."},{"Start":"01:08.705 ","End":"01:11.528","Text":"We get absolute value of 1^2 over 2,"},{"Start":"01:11.528 ","End":"01:15.350","Text":"and then the sum of all these coefficients squared."},{"Start":"01:15.350 ","End":"01:19.050","Text":"Remember that f is given as follows, 1, 0."},{"Start":"01:19.050 ","End":"01:25.295","Text":"This integral, this is just the integral from 0 to Pi,"},{"Start":"01:25.295 ","End":"01:26.540","Text":"because less than 0,"},{"Start":"01:26.540 ","End":"01:30.045","Text":"it\u0027s 0 of 1dx."},{"Start":"01:30.045 ","End":"01:34.815","Text":"This integral is Pi, but divided by Pi comes out to be 1."},{"Start":"01:34.815 ","End":"01:36.630","Text":"This is a 1/2,"},{"Start":"01:36.630 ","End":"01:40.250","Text":"and we can drop the absolute value and just square things,"},{"Start":"01:40.250 ","End":"01:44.695","Text":"4 over Pi squared 2k minus 1 squared. This is what we have now."},{"Start":"01:44.695 ","End":"01:47.170","Text":"Bring the 1/2 over to the other side."},{"Start":"01:47.170 ","End":"01:49.975","Text":"This series is equal to a half."},{"Start":"01:49.975 ","End":"01:54.430","Text":"Then we can divide by 4 and bring the Pi squared over,"},{"Start":"01:54.430 ","End":"01:56.715","Text":"so we get Pi squared over 8."},{"Start":"01:56.715 ","End":"02:01.330","Text":"This is what we had to show. We are done."}],"ID":28726},{"Watched":false,"Name":"Generalised Real Parseval Identity","Duration":"6m 33s","ChapterTopicVideoID":27539,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.810","Text":"Now we\u0027re continuing with the real case of Parseval\u0027s identity,"},{"Start":"00:03.810 ","End":"00:08.354","Text":"there is actually a generalized form and it goes as follows."},{"Start":"00:08.354 ","End":"00:10.109","Text":"Suppose we have 2 functions,"},{"Start":"00:10.109 ","End":"00:13.260","Text":"f and g on the interval minus Pi,"},{"Start":"00:13.260 ","End":"00:21.060","Text":"Pi and f has Fourier series as usual and g also,"},{"Start":"00:21.060 ","End":"00:27.540","Text":"but to distinguish, we\u0027ll use A and B for the coefficients."},{"Start":"00:27.540 ","End":"00:34.965","Text":"The generalized possible identity is that the integral of f g bar,"},{"Start":"00:34.965 ","End":"00:38.505","Text":"meaning g conjugate, is the following sum."},{"Start":"00:38.505 ","End":"00:42.040","Text":"Now, why is this a generalized form?"},{"Start":"00:42.040 ","End":"00:47.300","Text":"Because if we let f equal g, then on the left,"},{"Start":"00:47.300 ","End":"00:49.550","Text":"f g bar is f, f bar,"},{"Start":"00:49.550 ","End":"00:51.800","Text":"which is the absolute value of f squared,"},{"Start":"00:51.800 ","End":"00:53.780","Text":"just like in the regular one."},{"Start":"00:53.780 ","End":"00:57.140","Text":"Also here, all these coefficients."},{"Start":"00:57.140 ","End":"01:01.630","Text":"For example, this one will just be a_n bar,"},{"Start":"01:01.630 ","End":"01:03.695","Text":"which is absolute value of a_n squared,"},{"Start":"01:03.695 ","End":"01:06.650","Text":"so you see that even when f is equal to g it comes"},{"Start":"01:06.650 ","End":"01:09.529","Text":"out to be the regular Parseval\u0027s identity,"},{"Start":"01:09.529 ","End":"01:11.485","Text":"not the generalized one."},{"Start":"01:11.485 ","End":"01:14.630","Text":"Another remark, you previously encountered,"},{"Start":"01:14.630 ","End":"01:18.320","Text":"generalized Parseval\u0027s identity in the product spaces."},{"Start":"01:18.320 ","End":"01:24.965","Text":"I\u0027ve just mention it that this is a special case of the following in our space,"},{"Start":"01:24.965 ","End":"01:28.850","Text":"L^2_ PC won\u0027t go into that in more detail."},{"Start":"01:28.850 ","End":"01:33.725","Text":"Let\u0027s do an example of this to help clarify the concept."},{"Start":"01:33.725 ","End":"01:38.450","Text":"Let\u0027s take f to be the function which is defined here."},{"Start":"01:38.450 ","End":"01:39.770","Text":"I think I have a picture."},{"Start":"01:39.770 ","End":"01:41.285","Text":"Yeah, there it is."},{"Start":"01:41.285 ","End":"01:46.010","Text":"It\u0027s equal to x+Pi when x is"},{"Start":"01:46.010 ","End":"01:53.070","Text":"negative and just 0 when x is positive in this interval from minus Pi to Pi."},{"Start":"01:53.120 ","End":"01:57.485","Text":"I\u0027m just copying. This is the generalized Parseval\u0027s identity."},{"Start":"01:57.485 ","End":"01:59.390","Text":"What does it mean in our case?"},{"Start":"01:59.390 ","End":"02:05.075","Text":"G (x) from a previous exercise is equal to this series."},{"Start":"02:05.075 ","End":"02:08.360","Text":"Well, let\u0027s compute f(x)."},{"Start":"02:08.360 ","End":"02:09.530","Text":"F(x) is new to us,"},{"Start":"02:09.530 ","End":"02:12.410","Text":"so we need to compute a_n and b_n."},{"Start":"02:12.410 ","End":"02:15.635","Text":"Usually a_0 is a separate computation."},{"Start":"02:15.635 ","End":"02:20.520","Text":"Let\u0027s start with a_0 and after that,"},{"Start":"02:20.520 ","End":"02:23.865","Text":"we\u0027ll compute a_n and b_n."},{"Start":"02:23.865 ","End":"02:26.040","Text":"A_0 is 1 over Pi,"},{"Start":"02:26.040 ","End":"02:28.305","Text":"the integral of f(x)dx."},{"Start":"02:28.305 ","End":"02:30.690","Text":"That is equal to 1 over Pi."},{"Start":"02:30.690 ","End":"02:34.890","Text":"We don\u0027t need the part from 0 to Pi because that\u0027s 0,"},{"Start":"02:34.890 ","End":"02:37.290","Text":"and from minus Pi- to 0,"},{"Start":"02:37.290 ","End":"02:38.895","Text":"it\u0027s just equal to x plus Pi."},{"Start":"02:38.895 ","End":"02:40.945","Text":"This is the integral we get."},{"Start":"02:40.945 ","End":"02:44.735","Text":"I\u0027ll skip a couple of steps is very straightforward."},{"Start":"02:44.735 ","End":"02:47.165","Text":"The answer comes out to be Pi/2."},{"Start":"02:47.165 ","End":"02:49.775","Text":"Next we\u0027ll compute a_n,"},{"Start":"02:49.775 ","End":"02:51.560","Text":"this is the formula."},{"Start":"02:51.560 ","End":"02:53.120","Text":"Applying it to our case,"},{"Start":"02:53.120 ","End":"02:57.395","Text":"we get the integral from minus Pi to 0 of x plus Pi cosine nx."},{"Start":"02:57.395 ","End":"03:05.630","Text":"Then the integral of cosine nx is sine nx over n. This\u0027s by parts."},{"Start":"03:05.630 ","End":"03:09.680","Text":"Get the integral of minus sine, which is cosine,"},{"Start":"03:09.680 ","End":"03:13.280","Text":"and plug in the limits of integration and we"},{"Start":"03:13.280 ","End":"03:17.075","Text":"get this and we know that cosine of minus n Pi,"},{"Start":"03:17.075 ","End":"03:19.145","Text":"or you could throw out the minus,"},{"Start":"03:19.145 ","End":"03:23.090","Text":"because cosine is an even function and cosine n Pi is"},{"Start":"03:23.090 ","End":"03:26.965","Text":"minus 1 to the n. Now we can simplify this."},{"Start":"03:26.965 ","End":"03:29.085","Text":"Note, if n is even,"},{"Start":"03:29.085 ","End":"03:31.590","Text":"then we get 1 minus 1, 0."},{"Start":"03:31.590 ","End":"03:35.810","Text":"So y_n is 0 for even n and when n is odd,"},{"Start":"03:35.810 ","End":"03:38.845","Text":"we get 1 minus minus 1, which is 2."},{"Start":"03:38.845 ","End":"03:46.880","Text":"We get that a_n is 0 for even n and 2/Pi n^2."},{"Start":"03:46.880 ","End":"03:49.175","Text":"But n is 2k minus 1,"},{"Start":"03:49.175 ","End":"03:53.060","Text":"which is a general odd number When n is odd, we get this."},{"Start":"03:53.060 ","End":"03:55.810","Text":"Now let\u0027s go and find b_n."},{"Start":"03:55.810 ","End":"03:57.965","Text":"Just to save space,"},{"Start":"03:57.965 ","End":"04:00.770","Text":"put it in separate column here,"},{"Start":"04:00.770 ","End":"04:04.760","Text":"so b_n is this instead of cosine we have sine,"},{"Start":"04:04.760 ","End":"04:08.090","Text":"and it\u0027s almost exactly the same thing,"},{"Start":"04:08.090 ","End":"04:11.940","Text":"so I won\u0027t spend time on all the details of"},{"Start":"04:11.940 ","End":"04:16.130","Text":"the integrals to integration by parts, just very straightforward."},{"Start":"04:16.130 ","End":"04:20.900","Text":"We get that b_n is minus 1 over n. Now we have"},{"Start":"04:20.900 ","End":"04:27.500","Text":"b_n and we have a_n and earlier we got a_0."},{"Start":"04:27.500 ","End":"04:32.450","Text":"From these, we can piece it all together and put it in the formula"},{"Start":"04:32.450 ","End":"04:38.150","Text":"for the Fourier series representation of a function."},{"Start":"04:38.150 ","End":"04:45.020","Text":"It\u0027s a_0/2 plus the sum of a_n cosine nx."},{"Start":"04:45.020 ","End":"04:48.685","Text":"But we only need n which is equal to 2k minus 1."},{"Start":"04:48.685 ","End":"04:54.675","Text":"Here we need all the n from 1 to infinity minus 1 over n sine nx."},{"Start":"04:54.675 ","End":"04:57.915","Text":"G well, we have that already is this,"},{"Start":"04:57.915 ","End":"05:03.920","Text":"so now we\u0027re ready to plug this into the generalized Parseval identity."},{"Start":"05:03.920 ","End":"05:11.145","Text":"What we get that the integral 1 over Pi of f times g,"},{"Start":"05:11.145 ","End":"05:16.524","Text":"and we only need it from minus Pi to 0 is equal to a_0,"},{"Start":"05:16.524 ","End":"05:20.945","Text":"A_0 and then a_n, A_n."},{"Start":"05:20.945 ","End":"05:23.810","Text":"Anyway, we just substitute it, b_n,"},{"Start":"05:23.810 ","End":"05:27.230","Text":"B_n and then we can simplify this."},{"Start":"05:27.230 ","End":"05:32.836","Text":"Here we have x squared plus Pi x and here this drops off,"},{"Start":"05:32.836 ","End":"05:35.000","Text":"this drops off we\u0027re just left with this,"},{"Start":"05:35.000 ","End":"05:36.800","Text":"the minus and the minus cancel."},{"Start":"05:36.800 ","End":"05:41.750","Text":"We have minus 1 to the n over n. The 2 we pull out in front,"},{"Start":"05:41.750 ","End":"05:44.115","Text":"divide both sides by 2."},{"Start":"05:44.115 ","End":"05:45.770","Text":"We put the 2 here."},{"Start":"05:45.770 ","End":"05:49.880","Text":"This integral is x cubed over 3 plus Pi x squared over 2."},{"Start":"05:49.880 ","End":"05:55.245","Text":"Plug in the limits of integration and just switch left and right side."},{"Start":"05:55.245 ","End":"05:59.090","Text":"We get that this is equal to plug-in 0, we have 0."},{"Start":"05:59.090 ","End":"06:00.560","Text":"Plug in minus Pi,"},{"Start":"06:00.560 ","End":"06:05.420","Text":"we have this simplifies to minus Pi squared over 12."},{"Start":"06:05.420 ","End":"06:07.355","Text":"Notice we have a minus here."},{"Start":"06:07.355 ","End":"06:09.140","Text":"On the other hand, this series,"},{"Start":"06:09.140 ","End":"06:11.270","Text":"when n is 1, starts with a minus."},{"Start":"06:11.270 ","End":"06:13.355","Text":"We have minus plus minus plus."},{"Start":"06:13.355 ","End":"06:15.440","Text":"If we changed it to plus, minus,"},{"Start":"06:15.440 ","End":"06:19.340","Text":"plus minus, we can get rid of the minus here."},{"Start":"06:19.340 ","End":"06:25.885","Text":"What we\u0027re left with is that Pi squared over 12 is equal to the following,"},{"Start":"06:25.885 ","End":"06:30.905","Text":"the sum of alternating sum times 1 over n squared."},{"Start":"06:30.905 ","End":"06:34.770","Text":"So this is the answer and we are done."}],"ID":28727},{"Watched":false,"Name":"Exercise 2","Duration":"6m 22s","ChapterTopicVideoID":27546,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.295","Text":"The purpose of this exercise is to prove the identity."},{"Start":"00:05.295 ","End":"00:09.380","Text":"This infinite series sums to Pi^2 over 64."},{"Start":"00:09.380 ","End":"00:12.960","Text":"First we want to compute the real Fourier series of"},{"Start":"00:12.960 ","End":"00:17.510","Text":"this function and we have to assume that p is not an even number,"},{"Start":"00:17.510 ","End":"00:21.390","Text":"and using this we\u0027ll prove what we want to prove."},{"Start":"00:21.390 ","End":"00:28.250","Text":"Let\u0027s start with Part A note that f is an odd function because sin is an odd function."},{"Start":"00:28.250 ","End":"00:29.810","Text":"Here replace x, by minus x,"},{"Start":"00:29.810 ","End":"00:31.690","Text":"we just get a minus in front,"},{"Start":"00:31.690 ","End":"00:36.110","Text":"so that means that all the a n\u0027s in the series are 0s,"},{"Start":"00:36.110 ","End":"00:40.400","Text":"so we just have f as the sum of b_n sin nx."},{"Start":"00:40.400 ","End":"00:42.560","Text":"Now we have a formula for b_n,"},{"Start":"00:42.560 ","End":"00:43.990","Text":"which is this,"},{"Start":"00:43.990 ","End":"00:46.580","Text":"and note that sin is an odd function,"},{"Start":"00:46.580 ","End":"00:50.465","Text":"f is an odd function and an odd times an odd function is an even function,"},{"Start":"00:50.465 ","End":"00:52.850","Text":"so we can replace this integral on"},{"Start":"00:52.850 ","End":"00:57.995","Text":"the symmetric interval by twice the integral from 0 to Pi."},{"Start":"00:57.995 ","End":"01:01.280","Text":"Now replace f by what it\u0027s defined to be,"},{"Start":"01:01.280 ","End":"01:03.635","Text":"which is sin px over 2,"},{"Start":"01:03.635 ","End":"01:05.990","Text":"and we have sin something,"},{"Start":"01:05.990 ","End":"01:10.315","Text":"so I will use this trigonometric identity for product of sin\u0027s."},{"Start":"01:10.315 ","End":"01:11.830","Text":"If we do this,"},{"Start":"01:11.830 ","End":"01:16.870","Text":"the 2 here and the 2 here cancel each other out,"},{"Start":"01:16.870 ","End":"01:21.730","Text":"so we have 1 and then we have the cos of this minus this,"},{"Start":"01:21.730 ","End":"01:27.955","Text":"just taking the x out the bracket and similarly the cos of this plus this."},{"Start":"01:27.955 ","End":"01:31.480","Text":"Now the integral of cos is sin,"},{"Start":"01:31.480 ","End":"01:34.360","Text":"but we have to divide by the inner derivative here,"},{"Start":"01:34.360 ","End":"01:38.230","Text":"p over 2 minus n. Here we have to divide by p over 2 plus n,"},{"Start":"01:38.230 ","End":"01:40.615","Text":"and note that neither of these is 0,"},{"Start":"01:40.615 ","End":"01:43.870","Text":"because p over 2 is not a whole number,"},{"Start":"01:43.870 ","End":"01:46.745","Text":"we assume that p is not an even number."},{"Start":"01:46.745 ","End":"01:50.470","Text":"This gives us the following,"},{"Start":"01:50.470 ","End":"01:53.270","Text":"replace x equals Pi,"},{"Start":"01:53.270 ","End":"01:55.130","Text":"we have this and this,"},{"Start":"01:55.130 ","End":"02:00.110","Text":"and when x is 0, the sin of something times x is just 0."},{"Start":"02:00.110 ","End":"02:06.755","Text":"Sin (0) is 0 so that gives us the coefficient b_n or the coefficient."},{"Start":"02:06.755 ","End":"02:11.360","Text":"If we replace the b_ns in this formula,"},{"Start":"02:11.360 ","End":"02:18.005","Text":"what we get is this and we just have to simplify it a bit now."},{"Start":"02:18.005 ","End":"02:21.785","Text":"Here we are with this expression we want to simplify."},{"Start":"02:21.785 ","End":"02:26.135","Text":"Now, notice that this is a sin of something minus something."},{"Start":"02:26.135 ","End":"02:29.690","Text":"This is a sin of something plus something we\u0027re going to use this formula twice,"},{"Start":"02:29.690 ","End":"02:32.525","Text":"once in each of these so first let\u0027s do this 1."},{"Start":"02:32.525 ","End":"02:38.360","Text":"We get that the sin (p over 2 Pi minus n Pi) is sin,"},{"Start":"02:38.360 ","End":"02:41.125","Text":"cos minus cos sin,"},{"Start":"02:41.125 ","End":"02:44.190","Text":"and cos (nPi) is minus 1 to the n,"},{"Start":"02:44.190 ","End":"02:46.575","Text":"sin (nPi) is 0."},{"Start":"02:46.575 ","End":"02:51.940","Text":"This comes out to be minus 1 to the n sin (p over 2 Pi)."},{"Start":"02:51.940 ","End":"02:56.255","Text":"Similarly, almost exactly the same if we put a plus here,"},{"Start":"02:56.255 ","End":"02:58.055","Text":"we get the same thing."},{"Start":"02:58.055 ","End":"03:03.020","Text":"Because the plus becomes a plus here and we\u0027re just adding or subtracting 0,"},{"Start":"03:03.020 ","End":"03:04.550","Text":"so it\u0027s the same thing."},{"Start":"03:04.550 ","End":"03:12.830","Text":"Plug these 2 back in here and what we have is that f (x) as a Fourier series as follows."},{"Start":"03:12.830 ","End":"03:14.795","Text":"Now because these 2 are the same,"},{"Start":"03:14.795 ","End":"03:16.150","Text":"this is the same as this,"},{"Start":"03:16.150 ","End":"03:19.415","Text":"we can take it outside the brackets and"},{"Start":"03:19.415 ","End":"03:23.545","Text":"bring it in front so we now have 1 over p over 2 minus n,"},{"Start":"03:23.545 ","End":"03:27.755","Text":"minus 1 over p over 2 plus n and we can work on this,"},{"Start":"03:27.755 ","End":"03:33.925","Text":"cross-multiply p over 2 plus n takeaway p over 2 minus n is 2n,"},{"Start":"03:33.925 ","End":"03:39.740","Text":"and the product of the denominators differences of squares is p^2 over 4 minus n^2."},{"Start":"03:39.740 ","End":"03:42.650","Text":"Multiply top and bottom by 4 here,"},{"Start":"03:42.650 ","End":"03:44.390","Text":"so we have this,"},{"Start":"03:44.390 ","End":"03:50.145","Text":"we get that the Fourier series for sin px over 2 is the following,"},{"Start":"03:50.145 ","End":"03:52.860","Text":"and that\u0027s Part A."},{"Start":"03:52.860 ","End":"03:55.880","Text":"This is the result of Part A which we just showed."},{"Start":"03:55.880 ","End":"04:00.170","Text":"Now let\u0027s get on to Part B where we have to prove this identity."},{"Start":"04:00.170 ","End":"04:05.295","Text":"We apply Parseval\u0027s identity to the above,"},{"Start":"04:05.295 ","End":"04:08.585","Text":"this is Parseval\u0027s identity only in our case,"},{"Start":"04:08.585 ","End":"04:10.430","Text":"the a_n\u0027s are 0,"},{"Start":"04:10.430 ","End":"04:13.880","Text":"so we just have that this equals the sum of the absolute value"},{"Start":"04:13.880 ","End":"04:18.325","Text":"of (b_n)^2 and this comes out to be,"},{"Start":"04:18.325 ","End":"04:22.190","Text":"this, is this an absolute value squared."},{"Start":"04:22.190 ","End":"04:28.590","Text":"This is the b_n part,"},{"Start":"04:28.590 ","End":"04:33.034","Text":"that\u0027s here so just put that here in the absolute value squared."},{"Start":"04:33.034 ","End":"04:37.940","Text":"Now, this part is even so we can take from 0 to Pi and put a"},{"Start":"04:37.940 ","End":"04:43.025","Text":"2 here and here let\u0027s just square it let me throw out the absolute value."},{"Start":"04:43.025 ","End":"04:44.930","Text":"The h is the 64,"},{"Start":"04:44.930 ","End":"04:46.430","Text":"sin is sin^2,"},{"Start":"04:46.430 ","End":"04:48.965","Text":"the minus 1 disappears,"},{"Start":"04:48.965 ","End":"04:52.859","Text":"and the p over 2 times Pi,"},{"Start":"04:52.859 ","End":"04:56.105","Text":"it doesn\u0027t involve n, so we can put it in front of the sigma."},{"Start":"04:56.105 ","End":"04:59.935","Text":"We get this if we let p = 1 at this point,"},{"Start":"04:59.935 ","End":"05:03.020","Text":"we make the substitution,1 is not an even number,"},{"Start":"05:03.020 ","End":"05:04.330","Text":"so we can do that,"},{"Start":"05:04.330 ","End":"05:06.570","Text":"so this is what we get,"},{"Start":"05:06.570 ","End":"05:08.580","Text":"and p is 1,"},{"Start":"05:08.580 ","End":"05:12.150","Text":"so this is Pi over 2 and sin Pi over 2 is 1,"},{"Start":"05:12.150 ","End":"05:14.025","Text":"so sin^2 Pi over 2 is 1."},{"Start":"05:14.025 ","End":"05:16.755","Text":"This part can just be thrown out,"},{"Start":"05:16.755 ","End":"05:20.870","Text":"what we can do is take this 64 over Pi^2,"},{"Start":"05:20.870 ","End":"05:22.430","Text":"put it on the other side,"},{"Start":"05:22.430 ","End":"05:23.510","Text":"and then switch sides."},{"Start":"05:23.510 ","End":"05:26.720","Text":"That\u0027s the Pi^2 over 64 here."},{"Start":"05:26.720 ","End":"05:29.460","Text":"The Pi with the Pi cancels,"},{"Start":"05:29.460 ","End":"05:33.020","Text":"2 into 64 goes 32 so we have this,"},{"Start":"05:33.020 ","End":"05:37.495","Text":"now we want to do this integral we\u0027ll use this trigonometrical formula."},{"Start":"05:37.495 ","End":"05:40.649","Text":"The sin^2 (Alpha over 2) is this,"},{"Start":"05:40.649 ","End":"05:43.290","Text":"we\u0027ll take x as Alpha,"},{"Start":"05:43.290 ","End":"05:50.810","Text":"so we get 1 minus cos x and the 2 goes with the 32 to give 64 so we have this,"},{"Start":"05:50.810 ","End":"05:56.585","Text":"this is equal to the integral of 1 minus cos x is"},{"Start":"05:56.585 ","End":"06:03.245","Text":"x minus sin x and if we plug it in from 0 to Pi,"},{"Start":"06:03.245 ","End":"06:11.490","Text":"we get this, which is Pi^2 over 64 because sin Pi is 0, sin 0 is 0."},{"Start":"06:11.490 ","End":"06:13.710","Text":"We just have the Pi which goes with the Pi to make Pi^2"},{"Start":"06:13.710 ","End":"06:17.960","Text":"and so we have in our series = Pi^2 over 64,"},{"Start":"06:17.960 ","End":"06:19.160","Text":"which is what we wanted to show,"},{"Start":"06:19.160 ","End":"06:23.070","Text":"and that concludes Part B and this exercise."}],"ID":28728},{"Watched":false,"Name":"Exercise 3","Duration":"6m 7s","ChapterTopicVideoID":27533,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.030","Text":"In this exercise, we\u0027re going to compute the sum of this series."},{"Start":"00:06.030 ","End":"00:13.560","Text":"But first, we\u0027re going to compute the Fourier series of this function f defined here,"},{"Start":"00:13.560 ","End":"00:16.450","Text":"and this is the picture of it."},{"Start":"00:16.730 ","End":"00:19.610","Text":"Let\u0027s start straight away with a,"},{"Start":"00:19.610 ","End":"00:24.915","Text":"so f will have a Fourier series of the form, well,"},{"Start":"00:24.915 ","End":"00:31.170","Text":"as usual, and what we\u0027re going to do is compute the coefficients a_n and b_n,"},{"Start":"00:31.170 ","End":"00:34.695","Text":"and usually, a naught needs a separate calculation."},{"Start":"00:34.695 ","End":"00:40.170","Text":"As formulas for each of them b_n is the integral with the sign here,"},{"Start":"00:40.170 ","End":"00:42.960","Text":"a_n has the cosine here."},{"Start":"00:42.960 ","End":"00:45.925","Text":"I should have written it generically this,"},{"Start":"00:45.925 ","End":"00:53.104","Text":"and then in our particular case it comes down to this because of the way f is defined."},{"Start":"00:53.104 ","End":"00:57.110","Text":"We have 3 computations. Let\u0027s start with b_n."},{"Start":"00:57.110 ","End":"01:00.480","Text":"B_n it\u0027s going to be the integral"},{"Start":"01:00.480 ","End":"01:04.175","Text":"from h to Pi because that\u0027s the only place where it\u0027s non-zero,"},{"Start":"01:04.175 ","End":"01:07.955","Text":"and there it\u0027s h^2 sine (nx)dx."},{"Start":"01:07.955 ","End":"01:10.535","Text":"Integral of sine is cosine,"},{"Start":"01:10.535 ","End":"01:16.640","Text":"but we have to divide by n and then plug in h and plug in Pi,"},{"Start":"01:16.640 ","End":"01:19.775","Text":"reverse the order to get rid of the minus here."},{"Start":"01:19.775 ","End":"01:26.895","Text":"Then this comes out to be cosine of n_Pi is (-1)^n, and that\u0027s b_n."},{"Start":"01:26.895 ","End":"01:33.320","Text":"Now, a_n similar integral of cosine is sine."},{"Start":"01:33.320 ","End":"01:35.845","Text":"We\u0027ll take it from h to Pi,"},{"Start":"01:35.845 ","End":"01:39.420","Text":"so sine of n_Pi minus sine n_h,"},{"Start":"01:39.420 ","End":"01:41.700","Text":"but sine of n_Pi is 0,"},{"Start":"01:41.700 ","End":"01:43.110","Text":"bringing the minus in front,"},{"Start":"01:43.110 ","End":"01:51.615","Text":"and we have at a_n is minus h^2 over Pi sine n_h over n. That\u0027s for a naught."},{"Start":"01:51.615 ","End":"01:54.320","Text":"Well, h^2 comes out in front."},{"Start":"01:54.320 ","End":"01:56.630","Text":"This is the integral of 1 from h to Pi,"},{"Start":"01:56.630 ","End":"02:00.935","Text":"so it\u0027s just Pi minus h. Now we have b_n and a_n,"},{"Start":"02:00.935 ","End":"02:05.800","Text":"a node, and we have our formula for the Fourier series."},{"Start":"02:05.800 ","End":"02:11.934","Text":"Now we can plug each of these in a naught from here a_n from here,"},{"Start":"02:11.934 ","End":"02:14.230","Text":"and b_n from here and plug them in,"},{"Start":"02:14.230 ","End":"02:15.835","Text":"and this is what we get."},{"Start":"02:15.835 ","End":"02:18.040","Text":"This is the answer to Part A,"},{"Start":"02:18.040 ","End":"02:22.180","Text":"but let\u0027s remember these results that b_n is this,"},{"Start":"02:22.180 ","End":"02:24.970","Text":"a_n is this, and specifically,"},{"Start":"02:24.970 ","End":"02:27.770","Text":"a naught is this."},{"Start":"02:27.770 ","End":"02:32.935","Text":"I carried these formulas over for Part B where we have to,"},{"Start":"02:32.935 ","End":"02:35.500","Text":"forgot to say what Part B is."},{"Start":"02:35.500 ","End":"02:38.770","Text":"Here\u0027s a reminder what we wanted to compute."},{"Start":"02:38.770 ","End":"02:44.750","Text":"We\u0027re going to apply possibles identity to f(x),"},{"Start":"02:44.750 ","End":"02:50.115","Text":"and we need to plug in a naught a_n and b_n."},{"Start":"02:50.115 ","End":"02:51.885","Text":"We have all those here,"},{"Start":"02:51.885 ","End":"02:56.515","Text":"so the left-hand side is equal to this and this and this,"},{"Start":"02:56.515 ","End":"02:59.270","Text":"and we\u0027re going to work on both left and right-hand sides."},{"Start":"02:59.270 ","End":"03:03.305","Text":"On the left-hand side, we just replace f(x) by what it\u0027s equal,"},{"Start":"03:03.305 ","End":"03:06.995","Text":"which is h^2 but only from h to Pi."},{"Start":"03:06.995 ","End":"03:10.960","Text":"This thing squared is h^4 from h to Pi."},{"Start":"03:10.960 ","End":"03:16.490","Text":"Here, everywhere we can drop the absolute values because we\u0027re squaring and it\u0027s real,"},{"Start":"03:16.490 ","End":"03:22.650","Text":"so here we get h^2 over Pi^2 divided by 2, so it\u0027s this,"},{"Start":"03:22.650 ","End":"03:27.870","Text":"then we have Pi minus h^2 and here what we have in this term is"},{"Start":"03:27.870 ","End":"03:36.829","Text":"sine (n_h)^2 and this sine (n_h)^2 can be written as 1 minus cosine squared."},{"Start":"03:36.829 ","End":"03:40.655","Text":"That\u0027s the famous trigonometric identity, and here,"},{"Start":"03:40.655 ","End":"03:44.540","Text":"when we expand this squared and I written it in reverse order,"},{"Start":"03:44.540 ","End":"03:51.920","Text":"we get 1 minus twice this times this plus cosine (n_h)^2 and the thing"},{"Start":"03:51.920 ","End":"03:59.644","Text":"is that the cosine squared n_h will cancel because we have the same multiplier here."},{"Start":"03:59.644 ","End":"04:02.600","Text":"This one is the same as this one."},{"Start":"04:02.600 ","End":"04:06.725","Text":"When we combine, the cosine squared will cancel."},{"Start":"04:06.725 ","End":"04:09.605","Text":"On the right-hand side, we\u0027ll just get,"},{"Start":"04:09.605 ","End":"04:13.150","Text":"instead of this, we\u0027ll just get the 1,"},{"Start":"04:13.150 ","End":"04:15.860","Text":"and here when we get rid of that,"},{"Start":"04:15.860 ","End":"04:21.575","Text":"we\u0027ll just get 1 minus twice of this."},{"Start":"04:21.575 ","End":"04:25.530","Text":"That\u0027s it. On the left-hand side,"},{"Start":"04:25.530 ","End":"04:32.015","Text":"h^4 comes in front of the integral from h to Pi of 1 is Pi minus h. Now,"},{"Start":"04:32.015 ","End":"04:36.985","Text":"we can take this h_4 over Pi squared n^2 outside the brackets."},{"Start":"04:36.985 ","End":"04:43.325","Text":"In these two combined the sums and then we have 1+1 is 2."},{"Start":"04:43.325 ","End":"04:46.190","Text":"We bring the 2 out also because we have a 2 here,"},{"Start":"04:46.190 ","End":"04:50.235","Text":"so we have 2-2 times this."},{"Start":"04:50.235 ","End":"04:52.515","Text":"It\u0027s 1-1 times this,"},{"Start":"04:52.515 ","End":"04:55.895","Text":"and at this point we make a substitution."},{"Start":"04:55.895 ","End":"05:01.060","Text":"Let h=2, so h^4 is 16,"},{"Start":"05:01.060 ","End":"05:06.020","Text":"and here, h^4 over 2 is 8."},{"Start":"05:06.020 ","End":"05:09.660","Text":"H^4 is 16 times 2 is 32,"},{"Start":"05:09.660 ","End":"05:12.640","Text":"and here cosine n_h is cosine of 2n."},{"Start":"05:12.640 ","End":"05:21.380","Text":"Now multiply both sides by Pi squared over 32 to get rid of the 32 over Pi squared here,"},{"Start":"05:21.380 ","End":"05:26.360","Text":"and what we get if you multiply this by Pi squared over 32,"},{"Start":"05:26.360 ","End":"05:31.865","Text":"and here we get 16 into 32 goes twice the Pi squared over Pi is Pi,"},{"Start":"05:31.865 ","End":"05:33.590","Text":"and what do we have here?"},{"Start":"05:33.590 ","End":"05:35.730","Text":"Pi squared over 32,"},{"Start":"05:35.730 ","End":"05:39.800","Text":"Pi squared will cancel with Pi squared and 8 over 32 is a quarter."},{"Start":"05:39.800 ","End":"05:45.065","Text":"Here, this disappears, so this is what we have now, and next,"},{"Start":"05:45.065 ","End":"05:48.980","Text":"we bring this over to the other side, so it\u0027s a minus,"},{"Start":"05:48.980 ","End":"05:52.220","Text":"but then week switch left and right-hand side,"},{"Start":"05:52.220 ","End":"05:54.275","Text":"so we have this."},{"Start":"05:54.275 ","End":"05:57.020","Text":"The left-hand side is what we want to find."},{"Start":"05:57.020 ","End":"05:59.930","Text":"We just have to simplify the right-hand side a bit."},{"Start":"05:59.930 ","End":"06:03.710","Text":"Just straightforward algebra comes out to be Pi squared minus 4"},{"Start":"06:03.710 ","End":"06:08.280","Text":"over 4 and that\u0027s the answer to Part B and we are done."}],"ID":28729},{"Watched":false,"Name":"Complex Parseval Identity - Introduction","Duration":"4m 6s","ChapterTopicVideoID":27540,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"We just talked about the real case for Parseval\u0027s Identity."},{"Start":"00:04.650 ","End":"00:07.260","Text":"Now let\u0027s get onto the complex case."},{"Start":"00:07.260 ","End":"00:09.359","Text":"It\u0027s somewhat similar."},{"Start":"00:09.359 ","End":"00:13.050","Text":"We have f in the L_2 space of"},{"Start":"00:13.050 ","End":"00:19.185","Text":"piecewise continuous functions on the interval minus Pi, Pi complex function."},{"Start":"00:19.185 ","End":"00:24.545","Text":"Suppose it has the complex Fourier series as follows."},{"Start":"00:24.545 ","End":"00:30.635","Text":"Then what the theorem says is that this equality or identity holds."},{"Start":"00:30.635 ","End":"00:32.795","Text":"I won\u0027t read it out what it says,"},{"Start":"00:32.795 ","End":"00:34.700","Text":"it\u0027s like in the real case,"},{"Start":"00:34.700 ","End":"00:39.560","Text":"it\u0027s good amongst other things for finding certain infinite series."},{"Start":"00:39.560 ","End":"00:41.060","Text":"Let\u0027s do an example."},{"Start":"00:41.060 ","End":"00:47.820","Text":"Find the complex Fourier series of the function. This piece."},{"Start":"00:47.820 ","End":"00:50.220","Text":"I mean, this is true from 0 to 2pi,"},{"Start":"00:50.220 ","End":"00:54.275","Text":"but we wanted it from minus Pi to Pi."},{"Start":"00:54.275 ","End":"00:55.910","Text":"Here\u0027s the picture that could help."},{"Start":"00:55.910 ","End":"00:57.320","Text":"This is when we make it periodic."},{"Start":"00:57.320 ","End":"01:01.695","Text":"Otherwise we just take it from minus Pi to Pi."},{"Start":"01:01.695 ","End":"01:08.405","Text":"This portion here, from 0 to 2 Pi it makes not much difference."},{"Start":"01:08.405 ","End":"01:11.650","Text":"We have to use this series,"},{"Start":"01:11.650 ","End":"01:17.065","Text":"the Fourier series that we find to compute the following infinite sum."},{"Start":"01:17.065 ","End":"01:23.535","Text":"The Fourier series is of this form and we have a formula for CN, which is this."},{"Start":"01:23.535 ","End":"01:25.170","Text":"Let\u0027s compute that,"},{"Start":"01:25.170 ","End":"01:30.690","Text":"f(x) is 0 from minus Pi to 0 here."},{"Start":"01:30.690 ","End":"01:37.945","Text":"We can just drop that part and just take the integral from 0 to Pi, where it\u0027s 1."},{"Start":"01:37.945 ","End":"01:43.575","Text":"This is just the integral of e to the minus inx from 0 to Pi."},{"Start":"01:43.575 ","End":"01:45.320","Text":"This is the integral,"},{"Start":"01:45.320 ","End":"01:47.435","Text":"just divide it by minus in."},{"Start":"01:47.435 ","End":"01:49.250","Text":"This won\u0027t work if n is 0,"},{"Start":"01:49.250 ","End":"01:52.460","Text":"so we\u0027ll have to compute c_0 separately."},{"Start":"01:52.460 ","End":"01:56.505","Text":"See, this one comes out to be the one over 2pi,"},{"Start":"01:56.505 ","End":"02:01.690","Text":"then 1 minus e to the minus in Pi."},{"Start":"02:01.700 ","End":"02:04.370","Text":"E to the minus Pi,"},{"Start":"02:04.370 ","End":"02:05.900","Text":"Pi is minus 1,"},{"Start":"02:05.900 ","End":"02:09.635","Text":"so it\u0027s minus one to the n. This will be,"},{"Start":"02:09.635 ","End":"02:11.840","Text":"if n is even,"},{"Start":"02:11.840 ","End":"02:13.220","Text":"this will come out to be 1,"},{"Start":"02:13.220 ","End":"02:14.945","Text":"so 1 minus one is 0."},{"Start":"02:14.945 ","End":"02:16.475","Text":"If n is odd,"},{"Start":"02:16.475 ","End":"02:20.565","Text":"then we get 1 plus 1 over 2, which is 1."},{"Start":"02:20.565 ","End":"02:24.410","Text":"We get this when n is 2k plus 1."},{"Start":"02:24.410 ","End":"02:27.095","Text":"Now, like I said, we have to do C-naught separately."},{"Start":"02:27.095 ","End":"02:28.640","Text":"That\u0027s equal to 1 over 2pi,"},{"Start":"02:28.640 ","End":"02:37.380","Text":"the integral of 1 from 0 to Pi and e to the minus I-naught x is just 1."},{"Start":"02:37.380 ","End":"02:38.820","Text":"It\u0027s the integral of 1."},{"Start":"02:38.820 ","End":"02:43.220","Text":"It\u0027s Pi, Pi over 2pi comes out to be a half."},{"Start":"02:43.220 ","End":"02:47.300","Text":"Let\u0027s summarize. We have c_n and c_0 as follows."},{"Start":"02:47.300 ","End":"02:53.940","Text":"I just put this together and we get what f is as a complex Fourier series."},{"Start":"02:53.940 ","End":"02:55.850","Text":"The C_0 part is here,"},{"Start":"02:55.850 ","End":"02:57.680","Text":"and all the rest of it is here."},{"Start":"02:57.680 ","End":"03:00.110","Text":"This runs over all the odd numbers."},{"Start":"03:00.110 ","End":"03:04.385","Text":"Now time to apply the Parseval\u0027s identity."},{"Start":"03:04.385 ","End":"03:06.640","Text":"We get the following."},{"Start":"03:06.640 ","End":"03:14.610","Text":"What we get, remember f(x) is 1 only from 0 to Pi and 0 on the part from minus Pi to 0."},{"Start":"03:14.610 ","End":"03:16.700","Text":"We just have the integral from 0 to Pi."},{"Start":"03:16.700 ","End":"03:18.815","Text":"Absolute value of 1^2,"},{"Start":"03:18.815 ","End":"03:23.840","Text":"and this is equal to C naught is a half,"},{"Start":"03:23.840 ","End":"03:26.360","Text":"so we get absolute value of a half squared."},{"Start":"03:26.360 ","End":"03:28.580","Text":"All the rest of it, we only take the odd ones,"},{"Start":"03:28.580 ","End":"03:33.775","Text":"the 2k+1, and we get this part squared."},{"Start":"03:33.775 ","End":"03:37.400","Text":"All these positive drop the absolute value."},{"Start":"03:37.400 ","End":"03:44.719","Text":"Here we have 1 over 2pi and this integral is Pi This is a quarter plus this sum."},{"Start":"03:44.719 ","End":"03:46.955","Text":"Just a bit of algebra and I\u0027ll tidy up."},{"Start":"03:46.955 ","End":"03:54.215","Text":"Pi over 2pi is a half minus a quarter equals take the 1 over pi squared in front."},{"Start":"03:54.215 ","End":"03:59.630","Text":"Now this minus this is a quarter multiplied by Pi squared over 4 and switch"},{"Start":"03:59.630 ","End":"04:06.870","Text":"sides and we get that this is Pi squared over 4. We are done."}],"ID":28730},{"Watched":false,"Name":"Generalised Complex Parseval Identity","Duration":"2m 36s","ChapterTopicVideoID":27541,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.800","Text":"Now we\u0027re still with the complex version of the parts of our identity."},{"Start":"00:04.800 ","End":"00:09.600","Text":"But just like in the real case where we had a generalized form,"},{"Start":"00:09.600 ","End":"00:11.250","Text":"so in the complex case,"},{"Start":"00:11.250 ","End":"00:14.625","Text":"we have a generalized form of the identity."},{"Start":"00:14.625 ","End":"00:16.440","Text":"Here\u0027s what it says,"},{"Start":"00:16.440 ","End":"00:17.789","Text":"if we have 2 functions,"},{"Start":"00:17.789 ","End":"00:19.482","Text":"f and g,"},{"Start":"00:19.482 ","End":"00:22.575","Text":"with Fourier series as follows,"},{"Start":"00:22.575 ","End":"00:26.370","Text":"we\u0027ll just use the coefficients f_n and g_n to distinguish,"},{"Start":"00:26.370 ","End":"00:29.560","Text":"then 1 over 2Pi integral of f,"},{"Start":"00:29.560 ","End":"00:35.030","Text":"g conjugate is equal to the sum of f_n, g_n conjugate."},{"Start":"00:35.030 ","End":"00:38.120","Text":"The reason it\u0027s generalized is that"},{"Start":"00:38.120 ","End":"00:42.965","Text":"the other non-generalized case is a specific case of this where f=g."},{"Start":"00:42.965 ","End":"00:46.905","Text":"If f=g, we get absolute value of f^2 here,"},{"Start":"00:46.905 ","End":"00:50.203","Text":"and here we get absolute value of f_n^2."},{"Start":"00:50.203 ","End":"00:52.480","Text":"So it is a generalization."},{"Start":"00:52.480 ","End":"00:54.450","Text":"Let\u0027s do an example."},{"Start":"00:54.450 ","End":"00:59.396","Text":"We take f(x) to be 1 or e^(x^2),"},{"Start":"00:59.396 ","End":"01:03.455","Text":"depending on whether x is bigger than 0 or less than 0."},{"Start":"01:03.455 ","End":"01:05.450","Text":"That\u0027s this function here,"},{"Start":"01:05.450 ","End":"01:07.954","Text":"the green one bigger than 0,"},{"Start":"01:07.954 ","End":"01:09.830","Text":"it\u0027s the constant 1,"},{"Start":"01:09.830 ","End":"01:13.490","Text":"and less than 0, it\u0027s the function e^(x^2)."},{"Start":"01:13.490 ","End":"01:15.725","Text":"G, on the other hand,"},{"Start":"01:15.725 ","End":"01:20.565","Text":"is 0 except for x between 0 and 1,"},{"Start":"01:20.565 ","End":"01:23.275","Text":"where it\u0027s equal to 1 over x^2 plus 1,"},{"Start":"01:23.275 ","End":"01:24.420","Text":"and that\u0027s this bit here,"},{"Start":"01:24.420 ","End":"01:27.735","Text":"1 over x^2 plus 1, and 0 otherwise."},{"Start":"01:27.735 ","End":"01:34.419","Text":"Suppose the Fourier series are just like we assumed with f_n and g_n."},{"Start":"01:34.419 ","End":"01:37.560","Text":"Our task is to prove that the sum of f_n,"},{"Start":"01:37.560 ","End":"01:39.690","Text":"g_n bar is 1/8."},{"Start":"01:39.690 ","End":"01:44.810","Text":"Of course, we\u0027ll be using the generalized Parseval\u0027s identity for complex numbers,"},{"Start":"01:44.810 ","End":"01:46.265","Text":"and this is what it says."},{"Start":"01:46.265 ","End":"01:49.205","Text":"Let\u0027s work on the right-hand side."},{"Start":"01:49.205 ","End":"01:50.840","Text":"This is equal to,"},{"Start":"01:50.840 ","End":"01:54.275","Text":"it\u0027s only the bit between 0 and 1 that we need to consider."},{"Start":"01:54.275 ","End":"01:57.965","Text":"Because g is 0 outside of 0 and 1."},{"Start":"01:57.965 ","End":"02:00.170","Text":"We get the product of these 2."},{"Start":"02:00.170 ","End":"02:03.700","Text":"We get the 1 over 1 plus x^2 times the 1."},{"Start":"02:03.700 ","End":"02:08.785","Text":"It\u0027s just 1 over 1 plus x^2 between 0 and 1."},{"Start":"02:08.785 ","End":"02:11.269","Text":"This is a well-known integral."},{"Start":"02:11.269 ","End":"02:15.305","Text":"The integral of this is arctangent of x between 0 and 1."},{"Start":"02:15.305 ","End":"02:25.275","Text":"Now, the angle whose tangent is 0 is 0 and the angle whose tangent is 1 is Pi over 4."},{"Start":"02:25.275 ","End":"02:26.770","Text":"So we get 1 over 2Pi,"},{"Start":"02:26.770 ","End":"02:28.320","Text":"Pi over 4 minus 0,"},{"Start":"02:28.320 ","End":"02:31.380","Text":"and that comes out to be 1 over a,"},{"Start":"02:31.380 ","End":"02:36.640","Text":"the Pi cancels, 1/4 times 1/2 is an 1/8. We are done."}],"ID":28731},{"Watched":false,"Name":"Exercise 4 - Part a","Duration":"44s","ChapterTopicVideoID":27542,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"This exercise has 3 parts and we\u0027ll do each part in a separate clip."},{"Start":"00:04.890 ","End":"00:09.615","Text":"F is a 2 pi periodic function such that f(x)"},{"Start":"00:09.615 ","End":"00:15.495","Text":"is the sine of x for x between minus pi and pi."},{"Start":"00:15.495 ","End":"00:22.800","Text":"In part a, we have to sketch the graph of f on the interval from minus 3 pi to 3 pi."},{"Start":"00:22.800 ","End":"00:27.440","Text":"Now, 2 pi periodic means every 2 pi, it repeats itself."},{"Start":"00:27.440 ","End":"00:33.605","Text":"We start with sketching the sine from minus pi to pi,"},{"Start":"00:33.605 ","End":"00:36.109","Text":"and then just repeat."},{"Start":"00:36.109 ","End":"00:37.740","Text":"There\u0027s no need for more."},{"Start":"00:37.740 ","End":"00:39.000","Text":"It goes on to infinity,"},{"Start":"00:39.000 ","End":"00:41.270","Text":"but I think this gives the idea."},{"Start":"00:41.270 ","End":"00:44.220","Text":"That\u0027s all there is for part a."}],"ID":28732},{"Watched":false,"Name":"Exercise 4 - Part b","Duration":"2m 45s","ChapterTopicVideoID":27543,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.510","Text":"Now we come to Part B of this exercise just to remind you that f is defined as a"},{"Start":"00:06.510 ","End":"00:12.825","Text":"sine hyperbolic of x on the interval from minus Pi to Pi and then keeps repeating itself."},{"Start":"00:12.825 ","End":"00:16.725","Text":"You want to find the real Fourier series real as opposed to complex."},{"Start":"00:16.725 ","End":"00:19.965","Text":"This is just a technical exercise using the formulas."},{"Start":"00:19.965 ","End":"00:25.230","Text":"In general, the Fourier series will be what\u0027s written here."},{"Start":"00:25.230 ","End":"00:31.860","Text":"The reason I\u0027ve grayed out these is because sine hyperbolic is odd."},{"Start":"00:31.860 ","End":"00:34.470","Text":"Only the b_ns are non zero."},{"Start":"00:34.470 ","End":"00:37.260","Text":"The cosine parts are all 0."},{"Start":"00:37.260 ","End":"00:41.610","Text":"We have that sine hyperbolic x as Fourier series of the form b_n sine (nx),"},{"Start":"00:41.610 ","End":"00:43.905","Text":"we still have to find b_n."},{"Start":"00:43.905 ","End":"00:46.580","Text":"That\u0027s just use of the formula."},{"Start":"00:46.580 ","End":"00:48.830","Text":"We have to compute this integral."},{"Start":"00:48.830 ","End":"00:51.850","Text":"We\u0027ll do it with integration by parts."},{"Start":"00:51.850 ","End":"00:54.425","Text":"This is pretty standard."},{"Start":"00:54.425 ","End":"00:56.075","Text":"This is the formula."},{"Start":"00:56.075 ","End":"00:59.060","Text":"This is sine hyperbolic, that\u0027ll be f\u0027."},{"Start":"00:59.060 ","End":"01:02.030","Text":"The integral of sine hyperbolic is cosine"},{"Start":"01:02.030 ","End":"01:06.770","Text":"hyperbolic plus the derivative of sine is n cosine."},{"Start":"01:06.770 ","End":"01:12.920","Text":"This is 0, cosine n Pi is 0 and sine 0 is 0."},{"Start":"01:12.920 ","End":"01:14.690","Text":"We\u0027re left with this part."},{"Start":"01:14.690 ","End":"01:17.750","Text":"Then we\u0027ll do another integration by parts."},{"Start":"01:17.750 ","End":"01:19.875","Text":"This is f\u0027, this is g,"},{"Start":"01:19.875 ","End":"01:23.880","Text":"so f is sine hyperbolic and we get this."},{"Start":"01:23.880 ","End":"01:27.365","Text":"The derivative of this is minus n sine (nx)."},{"Start":"01:27.365 ","End":"01:31.000","Text":"This n, it comes out in front."},{"Start":"01:31.000 ","End":"01:35.970","Text":"This minus is from this minus here."},{"Start":"01:35.970 ","End":"01:39.200","Text":"Now, if we substitute 0,"},{"Start":"01:39.200 ","End":"01:41.210","Text":"sine hyperbolic is 0."},{"Start":"01:41.210 ","End":"01:42.860","Text":"If you substitute Pi,"},{"Start":"01:42.860 ","End":"01:46.640","Text":"we get sine hyperbolic of Pi and cosine of n Pi is minus"},{"Start":"01:46.640 ","End":"01:51.140","Text":"1 to the n. Here we get the minus n,"},{"Start":"01:51.140 ","End":"01:55.475","Text":"comes out front and with the minus becomes plus n,"},{"Start":"01:55.475 ","End":"01:57.665","Text":"then we have this integral."},{"Start":"01:57.665 ","End":"02:00.140","Text":"But it looks like we\u0027re going round in circles"},{"Start":"02:00.140 ","End":"02:03.500","Text":"because this is the integral we started off with."},{"Start":"02:03.500 ","End":"02:05.300","Text":"If we just keep doing it by parts,"},{"Start":"02:05.300 ","End":"02:06.845","Text":"we\u0027ll go round and round."},{"Start":"02:06.845 ","End":"02:11.000","Text":"The way to break the circle is to write an equation in b_n."},{"Start":"02:11.000 ","End":"02:15.805","Text":"This is b_n and we have b_n also on the right."},{"Start":"02:15.805 ","End":"02:21.375","Text":"We get b_n plus n^2 b_n when we bring this over."},{"Start":"02:21.375 ","End":"02:25.770","Text":"It\u0027s b_n 1 plus n^2 equals what\u0027s written here."},{"Start":"02:25.770 ","End":"02:28.035","Text":"Just some Algebra now,"},{"Start":"02:28.035 ","End":"02:33.095","Text":"divide both sides by 1 plus n^2 and just rearrange it a bit."},{"Start":"02:33.095 ","End":"02:36.335","Text":"Now plugged the b_n in the formula we had for b_n."},{"Start":"02:36.335 ","End":"02:41.880","Text":"You can see here this is b_n times sine (nx)."},{"Start":"02:41.880 ","End":"02:45.370","Text":"This is the answer. We are done."}],"ID":28733},{"Watched":false,"Name":"Exercise 4 - Part c","Duration":"2m 49s","ChapterTopicVideoID":27544,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.550","Text":"Now we come to Part C where we\u0027re asked to compute the sum of this series."},{"Start":"00:05.550 ","End":"00:09.435","Text":"The idea is to use the result of Part B."},{"Start":"00:09.435 ","End":"00:10.740","Text":"In Part B,"},{"Start":"00:10.740 ","End":"00:12.690","Text":"we got the Fourier series for f,"},{"Start":"00:12.690 ","End":"00:14.550","Text":"which is sine hyperbolic,"},{"Start":"00:14.550 ","End":"00:18.315","Text":"at least sine hyperbolic between minus Pi and Pi,"},{"Start":"00:18.315 ","End":"00:20.610","Text":"I should have repeated that."},{"Start":"00:20.610 ","End":"00:26.104","Text":"We\u0027ll apply the possible identity, the complex case."},{"Start":"00:26.104 ","End":"00:30.320","Text":"Now I\u0027ve crossed this out because we already showed that all the a_n"},{"Start":"00:30.320 ","End":"00:35.060","Text":"including a_0 are zero so we\u0027re just left with the b_n part."},{"Start":"00:35.060 ","End":"00:41.465","Text":"We get that this integral and didn\u0027t put the absolute value because it\u0027s real."},{"Start":"00:41.465 ","End":"00:44.845","Text":"When you square it, you don\u0027t need the absolute value."},{"Start":"00:44.845 ","End":"00:50.635","Text":"Here, just put the coefficient of the sine of nx here."},{"Start":"00:50.635 ","End":"00:56.135","Text":"Now, I\u0027ve colored the part that will just disappear because of the absolute value,"},{"Start":"00:56.135 ","End":"00:57.440","Text":"you don\u0027t need the minus here."},{"Start":"00:57.440 ","End":"00:59.920","Text":"We don\u0027t need the minus 1^n."},{"Start":"00:59.920 ","End":"01:04.325","Text":"We\u0027re going to use a trigonometric identity for the sine hyperbolic squared."},{"Start":"01:04.325 ","End":"01:09.320","Text":"The cosine hyperbolic of 2x is 2 sine hyperbolic squared plus 1."},{"Start":"01:09.320 ","End":"01:12.290","Text":"We could bring the 1 to the other side and then divide"},{"Start":"01:12.290 ","End":"01:15.700","Text":"by 2 to get a formula for sine hyperbolic squared."},{"Start":"01:15.700 ","End":"01:18.925","Text":"It\u0027s twice cosine hyperbolic 2x"},{"Start":"01:18.925 ","End":"01:23.030","Text":"minus 1 over 2 over 2 and the 2 put here is equal to this."},{"Start":"01:23.030 ","End":"01:25.100","Text":"After we thrown out the parts in red."},{"Start":"01:25.100 ","End":"01:29.750","Text":"I\u0027ve highlighted these in blue just for focus because we\u0027re aiming for this."},{"Start":"01:29.750 ","End":"01:32.365","Text":"What we have here is just this squared."},{"Start":"01:32.365 ","End":"01:35.900","Text":"Bring out all the other constants and squared,"},{"Start":"01:35.900 ","End":"01:37.700","Text":"we have sine hyperbolic squared."},{"Start":"01:37.700 ","End":"01:44.030","Text":"We have this 2 comes out as 4 and the Pi comes out as Pi squared times this sum."},{"Start":"01:44.030 ","End":"01:45.365","Text":"Also the integral here,"},{"Start":"01:45.365 ","End":"01:47.750","Text":"integral of cosine hyperbolic is sine hyperbolic,"},{"Start":"01:47.750 ","End":"01:49.520","Text":"but we have to divide by the 2."},{"Start":"01:49.520 ","End":"01:51.305","Text":"Integral of 1 is x."},{"Start":"01:51.305 ","End":"01:52.730","Text":"Now a bit of algebra,"},{"Start":"01:52.730 ","End":"01:58.135","Text":"first of all switch the sides and then divide both sides by this constant."},{"Start":"01:58.135 ","End":"02:01.910","Text":"It comes out as Pi squared in the numerator and"},{"Start":"02:01.910 ","End":"02:05.870","Text":"4 in the denominator and this in the denominator."},{"Start":"02:05.870 ","End":"02:08.240","Text":"The 4 with the 2 gives a."},{"Start":"02:08.240 ","End":"02:10.100","Text":"Each one of the Pi\u0027s cancels."},{"Start":"02:10.100 ","End":"02:11.765","Text":"Anyway, we get this."},{"Start":"02:11.765 ","End":"02:17.480","Text":"We still have to evaluate the integral because the sine hyperbolic is odd,"},{"Start":"02:17.480 ","End":"02:22.040","Text":"we can just evaluate from 0 to Pi and take it twice."},{"Start":"02:22.040 ","End":"02:26.695","Text":"The twice will cancel with the half and become 2x here."},{"Start":"02:26.695 ","End":"02:28.460","Text":"When x is 0,"},{"Start":"02:28.460 ","End":"02:29.720","Text":"both of these are zeros,"},{"Start":"02:29.720 ","End":"02:33.830","Text":"so we just have to substitute Pi and get sine hyperbolic 2Pi minus"},{"Start":"02:33.830 ","End":"02:39.830","Text":"2Pi together with the Pi over 8 sine hyperbolic squared of Pi."},{"Start":"02:39.830 ","End":"02:41.870","Text":"That\u0027s the answer."},{"Start":"02:41.870 ","End":"02:46.145","Text":"Approximately equal to this if you need a numerical value."},{"Start":"02:46.145 ","End":"02:49.890","Text":"That concludes this exercise."}],"ID":28734},{"Watched":false,"Name":"Exercise 5","Duration":"3m 18s","ChapterTopicVideoID":27545,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.645","Text":"In this exercise, we define a function f on the interval from minus Pi to Pi as follows."},{"Start":"00:06.645 ","End":"00:11.265","Text":"F(x) is the sum where n goes from 1 to infinity of the square root"},{"Start":"00:11.265 ","End":"00:17.250","Text":"of 1^2 minus 1 plus 2^2 times e^inx."},{"Start":"00:17.250 ","End":"00:22.200","Text":"Our task is to compute the integral from minus Pi to Pi,"},{"Start":"00:22.200 ","End":"00:26.835","Text":"absolute value of f(x) plus Pi minus f(x)^2 dx."},{"Start":"00:26.835 ","End":"00:30.000","Text":"We\u0027re going to use Parseval\u0027s identity."},{"Start":"00:30.000 ","End":"00:33.795","Text":"The first step will be to compute f(x) plus Pi."},{"Start":"00:33.795 ","End":"00:35.630","Text":"If we add Pi to x, well,"},{"Start":"00:35.630 ","End":"00:36.860","Text":"this doesn\u0027t contain x,"},{"Start":"00:36.860 ","End":"00:39.215","Text":"so we just have to add a Pi here."},{"Start":"00:39.215 ","End":"00:42.770","Text":"Then we\u0027ll use the property of exponents to express this as"},{"Start":"00:42.770 ","End":"00:47.855","Text":"a product and then e^in Pi is e^i Pi to the n,"},{"Start":"00:47.855 ","End":"00:50.135","Text":"which is minus 1^n."},{"Start":"00:50.135 ","End":"00:52.300","Text":"We can bring that in front."},{"Start":"00:52.300 ","End":"00:54.290","Text":"This is what we have."},{"Start":"00:54.290 ","End":"00:56.465","Text":"Then we\u0027re going to do the subtraction."},{"Start":"00:56.465 ","End":"00:59.240","Text":"We\u0027re working our way up to this integral in steps,"},{"Start":"00:59.240 ","End":"01:08.330","Text":"so the difference is minus 1 to the n minus 1 times this square root of e to the inx,"},{"Start":"01:08.330 ","End":"01:12.245","Text":"and then this is equal to this part."},{"Start":"01:12.245 ","End":"01:14.405","Text":"If n is even,"},{"Start":"01:14.405 ","End":"01:16.160","Text":"then it comes out to be 0,"},{"Start":"01:16.160 ","End":"01:18.740","Text":"it\u0027s 1 minus 1, and if n is odd,"},{"Start":"01:18.740 ","End":"01:20.360","Text":"this is minus 1, minus 1,"},{"Start":"01:20.360 ","End":"01:21.905","Text":"which is minus 2."},{"Start":"01:21.905 ","End":"01:27.485","Text":"We get this with a minus 2 or with a 0 depending on odd or even."},{"Start":"01:27.485 ","End":"01:33.060","Text":"What we can do is replace n by 2k minus 1 and leave out the zeros,"},{"Start":"01:33.060 ","End":"01:35.535","Text":"so we get this."},{"Start":"01:35.535 ","End":"01:41.095","Text":"This is of the form the sum c_n e^inx."},{"Start":"01:41.095 ","End":"01:49.655","Text":"Now Parseval\u0027s identity states that if f has a Fourier series complex of this form,"},{"Start":"01:49.655 ","End":"01:52.810","Text":"then this equality holds,"},{"Start":"01:52.810 ","End":"01:54.450","Text":"and we\u0027ll use this in a moment."},{"Start":"01:54.450 ","End":"01:56.150","Text":"First of all, let\u0027s do the last step."},{"Start":"01:56.150 ","End":"01:57.290","Text":"We have this difference,"},{"Start":"01:57.290 ","End":"02:00.155","Text":"now we want the integral of the absolute value squared,"},{"Start":"02:00.155 ","End":"02:03.180","Text":"so I\u0027ll put a 1/2Pi here."},{"Start":"02:03.180 ","End":"02:06.425","Text":"It will look like this when they compensate by putting a 2Pi."},{"Start":"02:06.425 ","End":"02:10.850","Text":"Then this integral of this difference squared just have to replace"},{"Start":"02:10.850 ","End":"02:15.080","Text":"this by the sum of the absolute value of c_n^2,"},{"Start":"02:15.080 ","End":"02:21.275","Text":"which is the sum of the absolute value of each of these squared."},{"Start":"02:21.275 ","End":"02:23.795","Text":"Now, the minus 2,"},{"Start":"02:23.795 ","End":"02:25.280","Text":"if you square it is 4,"},{"Start":"02:25.280 ","End":"02:27.755","Text":"the 4 with the 2 gives 8,"},{"Start":"02:27.755 ","End":"02:31.460","Text":"and the square root squared means we can just drop"},{"Start":"02:31.460 ","End":"02:35.165","Text":"the square root and put it in just regular square brackets."},{"Start":"02:35.165 ","End":"02:44.300","Text":"We get this. Now this series is a telescoping series. Let\u0027s see what we get."},{"Start":"02:44.300 ","End":"02:46.325","Text":"We get when k is 1,"},{"Start":"02:46.325 ","End":"02:48.860","Text":"we have 1/1^2 minus 1/3^2."},{"Start":"02:48.860 ","End":"02:52.625","Text":"Then when k is 2, we have 1/3^2 minus 1/5^2,"},{"Start":"02:52.625 ","End":"02:54.965","Text":"and 1/5^2 minus 1/7^2."},{"Start":"02:54.965 ","End":"03:00.980","Text":"Notice that each pair will cancel except for the first one and the last one,"},{"Start":"03:00.980 ","End":"03:02.060","Text":"but there is no last one,"},{"Start":"03:02.060 ","End":"03:07.285","Text":"so it all just cancels because the general term goes to 0 we\u0027re okay."},{"Start":"03:07.285 ","End":"03:11.805","Text":"We\u0027re just left here with 1 times 8Pi,"},{"Start":"03:11.805 ","End":"03:14.040","Text":"which is 8Pi,"},{"Start":"03:14.040 ","End":"03:19.120","Text":"and that\u0027s the answer to this exercise, and we are done."}],"ID":28735},{"Watched":false,"Name":"Exercise 6 - Part a","Duration":"4m 39s","ChapterTopicVideoID":27534,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:03.060","Text":"In this exercise, there are 3 parts,"},{"Start":"00:03.060 ","End":"00:10.830","Text":"but the main goal is to prove the identity in part c. But first we\u0027ll do a and b and a,"},{"Start":"00:10.830 ","End":"00:16.500","Text":"we have to find the complex Fourier series of f(x) equals sine x over 2 in part b,"},{"Start":"00:16.500 ","End":"00:19.620","Text":"we\u0027ll use this to prove that the sum from"},{"Start":"00:19.620 ","End":"00:25.350","Text":"minus infinity to infinity of this is pi^2 over 32."},{"Start":"00:25.350 ","End":"00:27.090","Text":"Then in part c,"},{"Start":"00:27.090 ","End":"00:33.045","Text":"we\u0027ll show that the sum just from 1 to infinity is Pi^2 over 64."},{"Start":"00:33.045 ","End":"00:35.865","Text":"I believe we\u0027ve shown this in another exercise,"},{"Start":"00:35.865 ","End":"00:38.220","Text":"but using the real Fourier series."},{"Start":"00:38.220 ","End":"00:40.260","Text":"If it looks familiar, you\u0027ll know why."},{"Start":"00:40.260 ","End":"00:42.725","Text":"Now let\u0027s start solving part a."},{"Start":"00:42.725 ","End":"00:46.700","Text":"The complex Fourier series in general has this form,"},{"Start":"00:46.700 ","End":"00:52.000","Text":"the sum of C_n e^inx and the sum is from minus infinity to infinity."},{"Start":"00:52.000 ","End":"00:54.430","Text":"We have a formula for C_n,"},{"Start":"00:54.430 ","End":"00:56.135","Text":"which is this formula,"},{"Start":"00:56.135 ","End":"01:00.440","Text":"which in our case comes out to be this integral."},{"Start":"01:00.440 ","End":"01:05.705","Text":"Now e^-inx can be written in terms of cosine and sine."},{"Start":"01:05.705 ","End":"01:11.600","Text":"This is basically Euler\u0027s formula with nx replacing Theta."},{"Start":"01:11.600 ","End":"01:16.775","Text":"What we get from this is the difference of 2 integrals."},{"Start":"01:16.775 ","End":"01:20.680","Text":"This minus i times this integral."},{"Start":"01:20.680 ","End":"01:25.280","Text":"Here we have an odd function times an even function which is odd."},{"Start":"01:25.280 ","End":"01:27.185","Text":"The integral will be 0."},{"Start":"01:27.185 ","End":"01:31.445","Text":"Here, the integral of an odd times odd equals even function."},{"Start":"01:31.445 ","End":"01:38.255","Text":"We can use the trick of doubling it and then just taking from 0-Pi."},{"Start":"01:38.255 ","End":"01:44.240","Text":"This part 0 here we have 2 times this integral."},{"Start":"01:44.240 ","End":"01:46.945","Text":"The 2 cancels out with this 2."},{"Start":"01:46.945 ","End":"01:50.630","Text":"We have an integral of sine times sine for that,"},{"Start":"01:50.630 ","End":"01:54.335","Text":"we\u0027ll use this formula for the product of sines."},{"Start":"01:54.335 ","End":"02:02.680","Text":"What that gives us here is sine and sine is the cosine of the difference."},{"Start":"02:02.680 ","End":"02:05.705","Text":"The 2 we\u0027ve put back in here,"},{"Start":"02:05.705 ","End":"02:08.240","Text":"the 2, first of all got canceled out."},{"Start":"02:08.240 ","End":"02:13.520","Text":"Now we\u0027re putting it back in so the cosine of x over 2 minus nx,"},{"Start":"02:13.520 ","End":"02:16.670","Text":"which we can take x out and here similarly,"},{"Start":"02:16.670 ","End":"02:21.585","Text":"x over 2 plus nx and we take the x out the brackets."},{"Start":"02:21.585 ","End":"02:24.065","Text":"Now we do the integral."},{"Start":"02:24.065 ","End":"02:26.780","Text":"The integral of cosine is sine,"},{"Start":"02:26.780 ","End":"02:29.000","Text":"but we have to divide by the inner derivative,"},{"Start":"02:29.000 ","End":"02:32.120","Text":"which is this here and this here."},{"Start":"02:32.120 ","End":"02:37.435","Text":"We have this and we just have to substitute the limits 0 with Pi and subtract."},{"Start":"02:37.435 ","End":"02:39.455","Text":"Now when x=0,"},{"Start":"02:39.455 ","End":"02:41.730","Text":"we gets sine 0 here and sine 0 here."},{"Start":"02:41.730 ","End":"02:45.935","Text":"So we can ignore that. We just have to substitute the Pi here."},{"Start":"02:45.935 ","End":"02:50.620","Text":"Instead of x, we have Pi, Pi."},{"Start":"02:50.620 ","End":"02:53.355","Text":"Now we want to simplify this."},{"Start":"02:53.355 ","End":"02:58.025","Text":"We\u0027ll use this trigonometric formula for the sine of a sum or difference."},{"Start":"02:58.025 ","End":"03:00.460","Text":"We use them in the numerators here."},{"Start":"03:00.460 ","End":"03:04.130","Text":"Here we have sine of Pi over 2 minus nPi,"},{"Start":"03:04.130 ","End":"03:06.095","Text":"so using this formula,"},{"Start":"03:06.095 ","End":"03:07.520","Text":"this is what we get."},{"Start":"03:07.520 ","End":"03:11.645","Text":"Similarly here using the formula with a plus, we\u0027ll get this."},{"Start":"03:11.645 ","End":"03:13.430","Text":"Now each of the pieces we can evaluate,"},{"Start":"03:13.430 ","End":"03:15.320","Text":"sine Pi over 2 is 1."},{"Start":"03:15.320 ","End":"03:18.955","Text":"Cosine of nPi is minus 1^n."},{"Start":"03:18.955 ","End":"03:22.680","Text":"Sine of nPi is 0."},{"Start":"03:22.680 ","End":"03:29.155","Text":"Similarly here, the same components just with the plus here."},{"Start":"03:29.155 ","End":"03:33.150","Text":"In the first term we just get 1 times minus"},{"Start":"03:33.150 ","End":"03:37.125","Text":"1^n is minus 1^n and this part doesn\u0027t contribute."},{"Start":"03:37.125 ","End":"03:39.965","Text":"Here also we have minus 1^n,"},{"Start":"03:39.965 ","End":"03:43.490","Text":"except that it\u0027s over denominator with a plus."},{"Start":"03:43.490 ","End":"03:48.605","Text":"Now, we can take the minus 1^n in front of the curly brackets."},{"Start":"03:48.605 ","End":"03:51.260","Text":"Also, the other thing we did here is multiply"},{"Start":"03:51.260 ","End":"03:54.595","Text":"this 2 in the denominator with the denominators here."},{"Start":"03:54.595 ","End":"03:57.945","Text":"We get 1 minus 2n, 1 plus 2n."},{"Start":"03:57.945 ","End":"04:00.000","Text":"If we do this subtraction,"},{"Start":"04:00.000 ","End":"04:03.840","Text":"we get 1 plus 2n minus 1 minus 2n,"},{"Start":"04:03.840 ","End":"04:06.960","Text":"which is 4n over a difference of squares."},{"Start":"04:06.960 ","End":"04:09.585","Text":"It\u0027s 1 minus 4n^2."},{"Start":"04:09.585 ","End":"04:11.360","Text":"Just rewrite this a bit,"},{"Start":"04:11.360 ","End":"04:14.155","Text":"put it all over 1 fraction."},{"Start":"04:14.155 ","End":"04:22.090","Text":"4 minus i and minus 1^n gives us this and Pi 1 minus 4n^2."},{"Start":"04:22.090 ","End":"04:23.540","Text":"That\u0027s not the end."},{"Start":"04:23.540 ","End":"04:25.280","Text":"Now that we have the C_n,"},{"Start":"04:25.280 ","End":"04:28.970","Text":"we can substitute in this formula for the sine,"},{"Start":"04:28.970 ","End":"04:32.515","Text":"for the Fourier series representation of sine x over 2."},{"Start":"04:32.515 ","End":"04:40.460","Text":"Just replace the C_n with this and we get this sum. That\u0027s part a."}],"ID":28736},{"Watched":false,"Name":"Exercise 6 - Part b","Duration":"2m 4s","ChapterTopicVideoID":27535,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"We just finished part a) of this exercise."},{"Start":"00:03.240 ","End":"00:05.835","Text":"Now let\u0027s do part b)."},{"Start":"00:05.835 ","End":"00:13.100","Text":"The result of part a) was that the Fourier series of sine x over 2 was given as follows;"},{"Start":"00:13.100 ","End":"00:15.810","Text":"this sum from minus infinity to infinity,"},{"Start":"00:15.810 ","End":"00:20.054","Text":"and we\u0027ll use Parseval\u0027s identity to prove this."},{"Start":"00:20.054 ","End":"00:24.570","Text":"Remember that the Parseval identity looks like this in the complex case."},{"Start":"00:24.570 ","End":"00:26.970","Text":"In our case, let\u0027s see what it gives us."},{"Start":"00:26.970 ","End":"00:30.630","Text":"I\u0027ll just copy this first and then instead have f(x),"},{"Start":"00:30.630 ","End":"00:32.790","Text":"put sine x over 2,"},{"Start":"00:32.790 ","End":"00:34.110","Text":"instead of C_n,"},{"Start":"00:34.110 ","End":"00:36.405","Text":"we put this co-efficient."},{"Start":"00:36.405 ","End":"00:37.950","Text":"Now inside the absolute value,"},{"Start":"00:37.950 ","End":"00:39.990","Text":"the minus doesn\u0027t make a difference,"},{"Start":"00:39.990 ","End":"00:44.030","Text":"the minus 1 to the n and the absolute value of I is 1."},{"Start":"00:44.030 ","End":"00:46.256","Text":"So you can get rid of those."},{"Start":"00:46.256 ","End":"00:48.860","Text":"Then we just have real number squared."},{"Start":"00:48.860 ","End":"00:50.870","Text":"We can drop the absolute value."},{"Start":"00:50.870 ","End":"00:55.595","Text":"Here also we can drop the absolute value because this is real, so this is what we get."},{"Start":"00:55.595 ","End":"00:59.330","Text":"Then we\u0027ll use this formula to do this integral."},{"Start":"00:59.330 ","End":"01:04.320","Text":"So sine squared x over 2 is 1 minus cosine x over 2."},{"Start":"01:04.320 ","End":"01:07.935","Text":"The 2 combines with the 2 here to give 4."},{"Start":"01:07.935 ","End":"01:11.150","Text":"On the right-hand side, it\u0027s the same thing."},{"Start":"01:11.150 ","End":"01:17.405","Text":"Now, let\u0027s switch sides and multiply both sides by Pi squared over 16,"},{"Start":"01:17.405 ","End":"01:19.670","Text":"the reciprocal of this."},{"Start":"01:19.670 ","End":"01:23.593","Text":"We have that this is this times Pi squared over 16."},{"Start":"01:23.593 ","End":"01:26.210","Text":"Something cancels here."},{"Start":"01:26.210 ","End":"01:31.610","Text":"We just get a single Pi and then the 16 and the 4 combine to give 64."},{"Start":"01:31.610 ","End":"01:35.050","Text":"This integral is 2 Pi,"},{"Start":"01:35.050 ","End":"01:40.970","Text":"because the integral of cosine from minus Pi to Pi is 0."},{"Start":"01:40.970 ","End":"01:46.430","Text":"Basically, you get sine of x and sine x is 0 both at Pi and minus Pi."},{"Start":"01:46.430 ","End":"01:50.410","Text":"So only the 1 contributes and the integral of 1 is 2 Pi."},{"Start":"01:50.410 ","End":"01:54.365","Text":"We have this, this is just Pi squared over 32,"},{"Start":"01:54.365 ","End":"01:55.610","Text":"and this is what we have to show."},{"Start":"01:55.610 ","End":"01:57.575","Text":"Let\u0027s look backward, scroll back."},{"Start":"01:57.575 ","End":"02:00.350","Text":"Yeah, here it is Pi squared over 32."},{"Start":"02:00.350 ","End":"02:02.195","Text":"We\u0027ve done part b),"},{"Start":"02:02.195 ","End":"02:04.800","Text":"and that\u0027s it for this clip."}],"ID":28737},{"Watched":false,"Name":"Exercise 6 - Part c","Duration":"2m 14s","ChapterTopicVideoID":27536,"CourseChapterTopicPlaylistID":294452,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"Continuing with this exercise,"},{"Start":"00:02.250 ","End":"00:06.390","Text":"we\u0027ve just done part b where we\u0027ve shown that the sum from minus infinity to"},{"Start":"00:06.390 ","End":"00:11.415","Text":"infinity of this series is Pi^2 over 32."},{"Start":"00:11.415 ","End":"00:16.590","Text":"We have to prove that if we take the sum from 1 to infinity,"},{"Start":"00:16.590 ","End":"00:19.590","Text":"then we get Pi^2 over 64."},{"Start":"00:19.590 ","End":"00:22.170","Text":"I\u0027ll give you a quick variable proof."},{"Start":"00:22.170 ","End":"00:25.410","Text":"If you put n equals 0 here,"},{"Start":"00:25.410 ","End":"00:26.910","Text":"we get 0,"},{"Start":"00:26.910 ","End":"00:32.955","Text":"so it\u0027s like having 2 sums from 1 to infinity and from minus 1 down to minus infinity."},{"Start":"00:32.955 ","End":"00:35.250","Text":"Now if we put n or minus n,"},{"Start":"00:35.250 ","End":"00:36.660","Text":"it\u0027s the same thing here,"},{"Start":"00:36.660 ","End":"00:38.510","Text":"so half of it is the positive part,"},{"Start":"00:38.510 ","End":"00:39.995","Text":"half of it is a negative part."},{"Start":"00:39.995 ","End":"00:43.535","Text":"We just have to divide this by 2 and get Pi^2 over 64."},{"Start":"00:43.535 ","End":"00:45.405","Text":"We have to do it properly now,"},{"Start":"00:45.405 ","End":"00:47.240","Text":"so we\u0027re starting with this,"},{"Start":"00:47.240 ","End":"00:50.780","Text":"the sum from minus infinity to infinity is Pi^2 over 32."},{"Start":"00:50.780 ","End":"00:53.420","Text":"Then we write it as three pieces."},{"Start":"00:53.420 ","End":"00:55.595","Text":"The sum from 1 to infinity,"},{"Start":"00:55.595 ","End":"01:01.835","Text":"just the 0 element and then the sum from minus infinity to minus 1."},{"Start":"01:01.835 ","End":"01:05.495","Text":"The sum of these three together is Pi^2 over 32."},{"Start":"01:05.495 ","End":"01:09.470","Text":"Different colors are this gray means that this drops out."},{"Start":"01:09.470 ","End":"01:12.590","Text":"We don\u0027t need this element is 0 and this one we\u0027re"},{"Start":"01:12.590 ","End":"01:16.085","Text":"going to rewrite to make it look like this one."},{"Start":"01:16.085 ","End":"01:20.870","Text":"What we\u0027ll do is make a substitution here."},{"Start":"01:20.870 ","End":"01:25.730","Text":"We\u0027ll replace n with minus n. So n minus"},{"Start":"01:25.730 ","End":"01:30.695","Text":"n here also n minus n and we have to substitute the limits."},{"Start":"01:30.695 ","End":"01:33.380","Text":"If n goes from minus infinity to minus 1,"},{"Start":"01:33.380 ","End":"01:37.580","Text":"minus n goes from 1 to infinity. This is what we get."},{"Start":"01:37.580 ","End":"01:39.845","Text":"Now, minus n^2 is n ^2,"},{"Start":"01:39.845 ","End":"01:42.170","Text":"minus n^2 again is n^2."},{"Start":"01:42.170 ","End":"01:46.430","Text":"Basically the general expression here stays the same,"},{"Start":"01:46.430 ","End":"01:48.860","Text":"but the sum instead of minus infinity to minus 1,"},{"Start":"01:48.860 ","End":"01:50.615","Text":"it goes from 1 to infinity,"},{"Start":"01:50.615 ","End":"01:56.030","Text":"which makes it the same as the first term in this trio."},{"Start":"01:56.030 ","End":"02:00.830","Text":"We have twice, this sum is Pi^2 over 32."},{"Start":"02:00.830 ","End":"02:04.475","Text":"Now all we have to do is to divide both sides by 2"},{"Start":"02:04.475 ","End":"02:08.975","Text":"and we get that 1 times the sum is Pi^2 over 64,"},{"Start":"02:08.975 ","End":"02:10.880","Text":"which is what we had to show."},{"Start":"02:10.880 ","End":"02:15.180","Text":"This completes part z and this whole exercise."}],"ID":28738}],"Thumbnail":null,"ID":294452},{"Name":"Riemann Lebesgue Lemma","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Riemann Lebesgue Lemma - Real Version","Duration":"1m 43s","ChapterTopicVideoID":27518,"CourseChapterTopicPlaylistID":294453,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"In this clip, we\u0027ll talk about"},{"Start":"00:02.460 ","End":"00:06.120","Text":"a small but useful result called the Riemann-Lebesgue Lemma."},{"Start":"00:06.120 ","End":"00:11.832","Text":"We\u0027ll do the real version here and in the following clip we\u0027ll do the complex version."},{"Start":"00:11.832 ","End":"00:14.520","Text":"It\u0027s based on the possible identities,"},{"Start":"00:14.520 ","End":"00:16.500","Text":"so I\u0027ll remind you what that is."},{"Start":"00:16.500 ","End":"00:17.700","Text":"Take a look at it,"},{"Start":"00:17.700 ","End":"00:19.275","Text":"this is what it is."},{"Start":"00:19.275 ","End":"00:22.430","Text":"If you look at this result,"},{"Start":"00:22.430 ","End":"00:28.415","Text":"these are both converging series so the general term has to go to 0."},{"Start":"00:28.415 ","End":"00:33.635","Text":"The limit as n goes to infinity of a_n is 0 and similarly for b_n."},{"Start":"00:33.635 ","End":"00:36.110","Text":"Might wonder about the absolute value;"},{"Start":"00:36.110 ","End":"00:39.050","Text":"I\u0027ll remind you that a sequence tends to"},{"Start":"00:39.050 ","End":"00:43.790","Text":"0 if and only if the absolute value also goes to 0."},{"Start":"00:43.790 ","End":"00:46.850","Text":"This simple conclusion is known as"},{"Start":"00:46.850 ","End":"00:50.680","Text":"the Riemann-Lebesgue Lemma and this is the real version."},{"Start":"00:50.680 ","End":"00:54.385","Text":"Let\u0027s do an example that uses this Lemma;"},{"Start":"00:54.385 ","End":"00:59.630","Text":"we have to compute the limit as n goes to infinity of this integral."},{"Start":"00:59.630 ","End":"01:06.840","Text":"Now this looks similar to the general formula for the b_n term in the Fourier expansion."},{"Start":"01:06.840 ","End":"01:11.240","Text":"In fact, if we let f(x) be this part here,"},{"Start":"01:11.240 ","End":"01:18.550","Text":"and first of all f is in the L2 space of piecewise functions and minus Pi, Pi."},{"Start":"01:18.550 ","End":"01:24.860","Text":"That means that we can use the formula here and write b_n as this."},{"Start":"01:24.860 ","End":"01:26.975","Text":"Now f(x) is this,"},{"Start":"01:26.975 ","End":"01:30.265","Text":"so we have the b_n as this."},{"Start":"01:30.265 ","End":"01:34.970","Text":"But the limit then goes to infinity of b_n is 0 and"},{"Start":"01:34.970 ","End":"01:40.475","Text":"so the limit of our integral is also 0 as n goes to infinity."},{"Start":"01:40.475 ","End":"01:43.740","Text":"That\u0027s all there is to it, we are done."}],"ID":28739},{"Watched":false,"Name":"Riemann Lebesgue Lemma - Complex Version","Duration":"2m 49s","ChapterTopicVideoID":27519,"CourseChapterTopicPlaylistID":294453,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:06.060","Text":"In this clip, we\u0027ll talk about the complex version of the Riemann Lebesgue Lemma."},{"Start":"00:06.060 ","End":"00:07.650","Text":"Just like in the real case,"},{"Start":"00:07.650 ","End":"00:10.140","Text":"it\u0027s based on possibles identity,"},{"Start":"00:10.140 ","End":"00:11.220","Text":"the complex case,"},{"Start":"00:11.220 ","End":"00:13.590","Text":"and this is here to remind you."},{"Start":"00:13.590 ","End":"00:17.400","Text":"One of the conclusions is that because this series"},{"Start":"00:17.400 ","End":"00:21.225","Text":"converges and it\u0027s from minus infinity to infinity,"},{"Start":"00:21.225 ","End":"00:23.655","Text":"the general term turns to 0,"},{"Start":"00:23.655 ","End":"00:27.765","Text":"both in the infinite direction and in the minus infinite direction."},{"Start":"00:27.765 ","End":"00:30.870","Text":"In other words, the limit as n goes to infinity of c_n is"},{"Start":"00:30.870 ","End":"00:35.339","Text":"0 and the limit of c_minus n is also 0."},{"Start":"00:35.339 ","End":"00:40.890","Text":"This conclusion is known as the Riemann Lebesgue Lemma complex version,"},{"Start":"00:40.890 ","End":"00:43.805","Text":"of course, That\u0027s basically it."},{"Start":"00:43.805 ","End":"00:46.495","Text":"Let\u0027s do an example that uses this."},{"Start":"00:46.495 ","End":"00:53.105","Text":"Our example is an exercise to compute the limit as n goes to infinity of this integral."},{"Start":"00:53.105 ","End":"00:55.820","Text":"Now we\u0027d like to use the formula for"},{"Start":"00:55.820 ","End":"00:59.060","Text":"the general term c_n and say that when n goes to infinity,"},{"Start":"00:59.060 ","End":"01:00.550","Text":"c_n goes to 0."},{"Start":"01:00.550 ","End":"01:02.490","Text":"But this is too different from this."},{"Start":"01:02.490 ","End":"01:04.590","Text":"Here, we don\u0027t just have a T,"},{"Start":"01:04.590 ","End":"01:07.355","Text":"we have n inside the integrand,"},{"Start":"01:07.355 ","End":"01:11.440","Text":"both as the square here and the limit is not from minus Pi to Pi."},{"Start":"01:11.440 ","End":"01:15.109","Text":"Lot of things wrong, but there\u0027s one thing that will fix all of these problems."},{"Start":"01:15.109 ","End":"01:17.905","Text":"If we just substitute x equals nt,"},{"Start":"01:17.905 ","End":"01:20.280","Text":"then dx is ndt,"},{"Start":"01:20.280 ","End":"01:23.570","Text":"and the limits of integration go from minus Pi over n,"},{"Start":"01:23.570 ","End":"01:26.075","Text":"Pi over n to minus Pi Pi."},{"Start":"01:26.075 ","End":"01:31.910","Text":"Then what we get after substituting is that this integral becomes this integral."},{"Start":"01:31.910 ","End":"01:34.305","Text":"The nt is the x,"},{"Start":"01:34.305 ","End":"01:37.290","Text":"then n and dx, sorry,"},{"Start":"01:37.290 ","End":"01:42.390","Text":"this should be dt not dx here and here."},{"Start":"01:42.390 ","End":"01:47.535","Text":"Continuing, ndt is dx,"},{"Start":"01:47.535 ","End":"01:51.200","Text":"and the limits of integration we said,"},{"Start":"01:51.200 ","End":"01:55.520","Text":"and also here n times t is x,"},{"Start":"01:55.520 ","End":"01:57.625","Text":"so we just have a single n left."},{"Start":"01:57.625 ","End":"02:02.865","Text":"Now we can define a function f(x) to be 1 over x^2 plus 1,"},{"Start":"02:02.865 ","End":"02:06.455","Text":"and then we can try using this formula."},{"Start":"02:06.455 ","End":"02:12.685","Text":"c_n is 1 over 2Pi the integral of f(x) e to the minus inx dx."},{"Start":"02:12.685 ","End":"02:16.910","Text":"It still doesn\u0027t look quite like what we have because we"},{"Start":"02:16.910 ","End":"02:20.830","Text":"have a plus here and here we have a minus with the i and x,"},{"Start":"02:20.830 ","End":"02:22.650","Text":"but not to worry."},{"Start":"02:22.650 ","End":"02:27.845","Text":"Recall that we can plug in instead of n minus n,"},{"Start":"02:27.845 ","End":"02:31.425","Text":"and then we\u0027ll get e to the inx here."},{"Start":"02:31.425 ","End":"02:36.920","Text":"The Riemann Lebesgue Lemma says that c_minus n also tends to 0,"},{"Start":"02:36.920 ","End":"02:39.205","Text":"c_n and c_minus n, both."},{"Start":"02:39.205 ","End":"02:41.250","Text":"C_minus n is this,"},{"Start":"02:41.250 ","End":"02:44.840","Text":"so we get that the limit of this integral is 0."},{"Start":"02:44.840 ","End":"02:46.790","Text":"That\u0027s the limit we wanted."},{"Start":"02:46.790 ","End":"02:49.890","Text":"The answer is 0 and we\u0027re done."}],"ID":28740},{"Watched":false,"Name":"Exercise 1","Duration":"4m 50s","ChapterTopicVideoID":27520,"CourseChapterTopicPlaylistID":294453,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.145","Text":"In this exercise, we have to prove that a certain limit is equal to 0."},{"Start":"00:05.145 ","End":"00:09.150","Text":"The limit of n times the integral of this"},{"Start":"00:09.150 ","End":"00:17.715","Text":"integral times e^inx dx at this limit is 0 and x is on the interval from minus Pi to Pi."},{"Start":"00:17.715 ","End":"00:21.600","Text":"This is actually a function of x,"},{"Start":"00:21.600 ","End":"00:22.620","Text":"call it f(x),"},{"Start":"00:22.620 ","End":"00:24.465","Text":"which I slightly colored."},{"Start":"00:24.465 ","End":"00:28.050","Text":"We want to prove that this limit is 0,"},{"Start":"00:28.050 ","End":"00:31.830","Text":"which comes out to be the limit of n times the integral from"},{"Start":"00:31.830 ","End":"00:36.885","Text":"minus Pi to Pi of f(x), e^inx dx."},{"Start":"00:36.885 ","End":"00:40.730","Text":"Let\u0027s work on this part without the limit and without the 0."},{"Start":"00:40.730 ","End":"00:42.470","Text":"We\u0027ll see what we can do with it."},{"Start":"00:42.470 ","End":"00:45.185","Text":"We can say that this is equal to,"},{"Start":"00:45.185 ","End":"00:50.705","Text":"I\u0027m going to use integration by parts to continue with this,"},{"Start":"00:50.705 ","End":"00:52.715","Text":"f(x) is just f(x),"},{"Start":"00:52.715 ","End":"00:55.720","Text":"and this will be our g prime of x."},{"Start":"00:55.720 ","End":"01:00.665","Text":"What we get is f(x) times g(x)."},{"Start":"01:00.665 ","End":"01:02.225","Text":"The integral of this,"},{"Start":"01:02.225 ","End":"01:07.715","Text":"we take e^inx times n and divide it by in."},{"Start":"01:07.715 ","End":"01:10.625","Text":"The n cancels and it\u0027s just divided by i,"},{"Start":"01:10.625 ","End":"01:16.000","Text":"minus the derivative of f also times g(x), which is this."},{"Start":"01:16.000 ","End":"01:21.260","Text":"The next step is to show that f is an even function,"},{"Start":"01:21.260 ","End":"01:22.310","Text":"that will help us."},{"Start":"01:22.310 ","End":"01:28.085","Text":"f(x) is this integral and we can make a substitution."},{"Start":"01:28.085 ","End":"01:33.810","Text":"We can let u equals s^2 here and then du is 2sds,"},{"Start":"01:33.810 ","End":"01:38.180","Text":"and the limits of integration will change to 0x^2."},{"Start":"01:38.180 ","End":"01:40.250","Text":"After the substitution,"},{"Start":"01:40.250 ","End":"01:41.900","Text":"this is what we get."},{"Start":"01:41.900 ","End":"01:44.450","Text":"The only place x appears is here."},{"Start":"01:44.450 ","End":"01:50.665","Text":"It\u0027s clear if we replace x by minus x that\u0027s not going to change."},{"Start":"01:50.665 ","End":"01:54.650","Text":"f(x) is f(-x) and f is an even function,"},{"Start":"01:54.650 ","End":"02:02.260","Text":"and in particular, what we need is that f(Pi) is f(-Pi)."},{"Start":"02:02.260 ","End":"02:07.820","Text":"There\u0027s an easier way to show that f is even without doing this substitution."},{"Start":"02:07.820 ","End":"02:09.185","Text":"I realized it afterwards."},{"Start":"02:09.185 ","End":"02:17.160","Text":"It\u0027s enough that this is an odd function of s. f is some odd function of s,"},{"Start":"02:17.160 ","End":"02:18.510","Text":"call it g(s),"},{"Start":"02:18.510 ","End":"02:21.750","Text":"and this is the oddness definition."},{"Start":"02:21.750 ","End":"02:25.700","Text":"Then if we take the integral from 0 to minus x,"},{"Start":"02:25.700 ","End":"02:27.350","Text":"which is f (-x),"},{"Start":"02:27.350 ","End":"02:32.045","Text":"we can make a substitution of s equals minus t and get this,"},{"Start":"02:32.045 ","End":"02:34.295","Text":"and this comes back to g(t),"},{"Start":"02:34.295 ","End":"02:38.580","Text":"so this integral doesn\u0027t matter if it\u0027s t or s. Comes out to be f(x)."},{"Start":"02:38.580 ","End":"02:40.710","Text":"f is an even function."},{"Start":"02:40.710 ","End":"02:42.120","Text":"We\u0027re back to this point,"},{"Start":"02:42.120 ","End":"02:44.940","Text":"and again, f(Pi) is f(-Pi)."},{"Start":"02:44.940 ","End":"02:51.950","Text":"Next, we\u0027ll use this fact to show that this part of the integral here is 0,"},{"Start":"02:51.950 ","End":"02:54.155","Text":"and then we\u0027ll just be left with this."},{"Start":"02:54.155 ","End":"02:55.490","Text":"Let\u0027s evaluate this."},{"Start":"02:55.490 ","End":"02:59.330","Text":"We need to substitute Pi and then minus Pi and subtract."},{"Start":"02:59.330 ","End":"03:01.020","Text":"We get, first of all,"},{"Start":"03:01.020 ","End":"03:03.540","Text":"f of Pi, e^in Pi over i,"},{"Start":"03:03.540 ","End":"03:05.565","Text":"and then f of minus Pi,"},{"Start":"03:05.565 ","End":"03:08.450","Text":"e^minus in Pi over i."},{"Start":"03:08.450 ","End":"03:15.620","Text":"This part is equal to this part because they\u0027re both minus 1 to the power of n,"},{"Start":"03:15.620 ","End":"03:18.200","Text":"e^iPi is minus 1,"},{"Start":"03:18.200 ","End":"03:20.645","Text":"e to the minus iPi minus 1."},{"Start":"03:20.645 ","End":"03:23.060","Text":"Also f(Pi) is f(-Pi)."},{"Start":"03:23.060 ","End":"03:24.745","Text":"That\u0027s what we\u0027ve just shown."},{"Start":"03:24.745 ","End":"03:27.075","Text":"This minus this is 0,"},{"Start":"03:27.075 ","End":"03:30.855","Text":"and now we\u0027re left with computing the second part."},{"Start":"03:30.855 ","End":"03:32.750","Text":"This was equal to this minus this,"},{"Start":"03:32.750 ","End":"03:35.780","Text":"this part 0. We\u0027ve got this."},{"Start":"03:35.780 ","End":"03:40.925","Text":"To simplify it, minus 1 over i is the same as i,"},{"Start":"03:40.925 ","End":"03:43.610","Text":"because i squared is minus 1."},{"Start":"03:43.610 ","End":"03:48.205","Text":"We can write the i as 2Pi i over 2Pi."},{"Start":"03:48.205 ","End":"03:51.230","Text":"Reason for this, is that this looks familiar."},{"Start":"03:51.230 ","End":"03:58.505","Text":"This in fact is a formula for C_minus n. I\u0027ll explain what I mean."},{"Start":"03:58.505 ","End":"04:02.915","Text":"f\u0027(x) has a Fourier expansion complex"},{"Start":"04:02.915 ","End":"04:06.860","Text":"as sum from minus infinity to infinity is C_n e^inx."},{"Start":"04:06.860 ","End":"04:09.560","Text":"We have a formula for C_n which is this,"},{"Start":"04:09.560 ","End":"04:11.870","Text":"which has a minus in the formula."},{"Start":"04:11.870 ","End":"04:14.555","Text":"But we have a plus in the formula."},{"Start":"04:14.555 ","End":"04:17.600","Text":"If you compute C_minus n,"},{"Start":"04:17.600 ","End":"04:19.475","Text":"it will be exactly this."},{"Start":"04:19.475 ","End":"04:21.710","Text":"What we have here, is C_minus n,"},{"Start":"04:21.710 ","End":"04:23.885","Text":"the minus n coefficient."},{"Start":"04:23.885 ","End":"04:29.630","Text":"The limit that we wanted is equal to the limit of 2Pi i,"},{"Start":"04:29.630 ","End":"04:34.520","Text":"C minus n. You can bring the 2Pi i in front and we have the limit of C_minus n,"},{"Start":"04:34.520 ","End":"04:36.710","Text":"and now is where we apply"},{"Start":"04:36.710 ","End":"04:43.120","Text":"Riemann–Lebesgue Lemma and we get that this is equal to 0 by that Lemma."},{"Start":"04:43.430 ","End":"04:48.320","Text":"We\u0027ve proven that this limit is equal to 0,"},{"Start":"04:48.320 ","End":"04:51.570","Text":"which is what we had to show and we are done."}],"ID":28741}],"Thumbnail":null,"ID":294453},{"Name":"Dirichlet\u0027s Theorem","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Dirichlet\u0026#39;s Theorem","Duration":"5m 49s","ChapterTopicVideoID":27560,"CourseChapterTopicPlaylistID":294454,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:05.320","Text":"Now we come to the Dirichlet\u0027s theorem on pointwise convergence."},{"Start":"00:05.320 ","End":"00:09.580","Text":"Now, when we write this tilde squiggly as in"},{"Start":"00:09.580 ","End":"00:14.830","Text":"f tilde and this series is not the same as equals."},{"Start":"00:14.830 ","End":"00:18.820","Text":"In some ways, f is represented by this series,"},{"Start":"00:18.820 ","End":"00:21.032","Text":"but it\u0027s not exactly equality,"},{"Start":"00:21.032 ","End":"00:23.755","Text":"but sometimes we do need equality."},{"Start":"00:23.755 ","End":"00:26.825","Text":"If we have really written f(x)."},{"Start":"00:26.825 ","End":"00:30.700","Text":"This is a number and this is a series of numbers."},{"Start":"00:30.700 ","End":"00:35.205","Text":"Once we have a specific x we want to know when we can write equals here."},{"Start":"00:35.205 ","End":"00:38.685","Text":"For that, we come to Dirichlet\u0027s theorem,"},{"Start":"00:38.685 ","End":"00:40.950","Text":"which gives us a condition."},{"Start":"00:40.950 ","End":"00:42.810","Text":"Here\u0027s what it is."},{"Start":"00:42.810 ","End":"00:47.790","Text":"If f is piecewise continuous on minus Pi, Pi,"},{"Start":"00:47.790 ","End":"00:50.180","Text":"and at some point in the open interval,"},{"Start":"00:50.180 ","End":"00:55.789","Text":"there exists both one sided derivatives from the right and from the left."},{"Start":"00:55.789 ","End":"01:02.375","Text":"Then the Fourier series at the point x note is equal to"},{"Start":"01:02.375 ","End":"01:09.155","Text":"the average of the right limit of f at x_0 and the left limit of f at x_0."},{"Start":"01:09.155 ","End":"01:11.360","Text":"It has one sided derivatives,"},{"Start":"01:11.360 ","End":"01:14.540","Text":"so it certainly has limits from the left and on the right."},{"Start":"01:14.540 ","End":"01:18.780","Text":"This is the average which you might expect. That\u0027s the real case."},{"Start":"01:18.780 ","End":"01:21.020","Text":"Similarly in the complex case,"},{"Start":"01:21.020 ","End":"01:24.020","Text":"the Fourier series converges to"},{"Start":"01:24.020 ","End":"01:29.300","Text":"the average of the right limit of f and the left limit of f at x_0."},{"Start":"01:29.300 ","End":"01:34.055","Text":"Now some remarks and then we\u0027ll do an example exercise. First remark."},{"Start":"01:34.055 ","End":"01:37.115","Text":"If f is continuous at x_0,"},{"Start":"01:37.115 ","End":"01:42.830","Text":"then we actually have equality that f of x_0 is the sum of the Fourier series,"},{"Start":"01:42.830 ","End":"01:45.580","Text":"whether it\u0027s real or whether it\u0027s complex in either case."},{"Start":"01:45.580 ","End":"01:48.890","Text":"The reason for that is that if f is continuous,"},{"Start":"01:48.890 ","End":"01:52.220","Text":"then the limit of f from the right"},{"Start":"01:52.220 ","End":"01:55.430","Text":"and the limit from the left to the x_0 are both the same and equal,"},{"Start":"01:55.430 ","End":"01:57.370","Text":"and they\u0027re equal to f of x_0."},{"Start":"01:57.370 ","End":"02:00.270","Text":"The average is f of x_0."},{"Start":"02:00.270 ","End":"02:03.740","Text":"Second point is that there\u0027s no requirement that f"},{"Start":"02:03.740 ","End":"02:07.865","Text":"should be differentiable of x_0, just note that."},{"Start":"02:07.865 ","End":"02:14.075","Text":"The third remark relates to the end points minus Pi and Pi."},{"Start":"02:14.075 ","End":"02:22.635","Text":"If there are the one sided derivatives from the left at Pi and from the right minus Pi,"},{"Start":"02:22.635 ","End":"02:26.385","Text":"then at each of the end points plus or minus Pi,"},{"Start":"02:26.385 ","End":"02:29.420","Text":"the Fourier series converges to the average of"},{"Start":"02:29.420 ","End":"02:34.685","Text":"these two limits minus Pi on the right and plus Pi on the left."},{"Start":"02:34.685 ","End":"02:38.915","Text":"This makes sense because if you extend f periodically,"},{"Start":"02:38.915 ","End":"02:45.315","Text":"then really Pi and minus Pi are the same in the continuing periodic extension."},{"Start":"02:45.315 ","End":"02:49.110","Text":"It\u0027s like at the same point Pi from the right and from the left."},{"Start":"02:49.110 ","End":"02:51.885","Text":"Now to the example exercise."},{"Start":"02:51.885 ","End":"02:58.670","Text":"Recall from a previous exercise that the Fourier series for f(x)=x on minus Pi,"},{"Start":"02:58.670 ","End":"03:01.820","Text":"Pi is the following Fourier series."},{"Start":"03:01.820 ","End":"03:03.650","Text":"It\u0027s made up of signs."},{"Start":"03:03.650 ","End":"03:07.550","Text":"You have to use this to prove a famous identity that"},{"Start":"03:07.550 ","End":"03:13.750","Text":"1/4(Pi) is one minus 1/3 plus 1/5 minus 1/7 plus 1/9 minus 1/11 and so on."},{"Start":"03:13.750 ","End":"03:19.885","Text":"We\u0027re given a hint to substitute x= Pi over 2 in the Fourier series."},{"Start":"03:19.885 ","End":"03:23.644","Text":"I copied the series for x."},{"Start":"03:23.644 ","End":"03:29.480","Text":"Now we can use Dirichlet\u0027s theorem because f is continuous at Pi over 2."},{"Start":"03:29.480 ","End":"03:30.965","Text":"Perhaps a picture will help."},{"Start":"03:30.965 ","End":"03:32.730","Text":"Pi over 2 is no problem,"},{"Start":"03:32.730 ","End":"03:38.760","Text":"is only at Pi and minus Pi is our discontinuity but Pi over 2 everything is continuous."},{"Start":"03:38.760 ","End":"03:40.075","Text":"By Dirichlet\u0027s theorem,"},{"Start":"03:40.075 ","End":"03:43.955","Text":"we\u0027re allowed to substitute Pi over 2 and get equality here."},{"Start":"03:43.955 ","End":"03:51.305","Text":"We have the Pi over 2 which is x equals the series with Pi over 2 in place of x."},{"Start":"03:51.305 ","End":"03:54.620","Text":"Now, notice that if n is even,"},{"Start":"03:54.620 ","End":"04:00.155","Text":"say n is 2k, then we get sine of k Pi, which is 0,"},{"Start":"04:00.155 ","End":"04:01.820","Text":"and if n is odd,"},{"Start":"04:01.820 ","End":"04:05.035","Text":"typical odd number is 2k minus 1,"},{"Start":"04:05.035 ","End":"04:10.735","Text":"then sine of n Pi over 2 is k Pi minus Pi over 2."},{"Start":"04:10.735 ","End":"04:13.820","Text":"We can switch the order and put a minus in front."},{"Start":"04:13.820 ","End":"04:15.980","Text":"Then using trig identities,"},{"Start":"04:15.980 ","End":"04:20.585","Text":"this is equal to minus the cosine of k Pi."},{"Start":"04:20.585 ","End":"04:23.045","Text":"We know what cosine of k Pi is."},{"Start":"04:23.045 ","End":"04:24.710","Text":"It\u0027s minus 1^n,"},{"Start":"04:24.710 ","End":"04:26.405","Text":"but we still have a minus here."},{"Start":"04:26.405 ","End":"04:28.070","Text":"We get minus 1,"},{"Start":"04:28.070 ","End":"04:31.255","Text":"I just raise the power by 1."},{"Start":"04:31.255 ","End":"04:33.930","Text":"There should be a k. You can figure that out."},{"Start":"04:33.930 ","End":"04:38.670","Text":"That\u0027s a k, not an n. Pi over 2 is the sum of"},{"Start":"04:38.670 ","End":"04:42.470","Text":"the series here but we only need to take the values of"},{"Start":"04:42.470 ","End":"04:47.390","Text":"n which are 2k minus 1 because the even ones comes out 0."},{"Start":"04:47.390 ","End":"04:49.850","Text":"Now, if n is 2k minus 1,"},{"Start":"04:49.850 ","End":"04:55.255","Text":"we saw here that sine of n Pi over 2 is minus 1^k plus 1."},{"Start":"04:55.255 ","End":"04:57.830","Text":"We get minus 1^2k,"},{"Start":"04:57.830 ","End":"05:01.440","Text":"because n is 2k minus 1 plus 1 is 2k."},{"Start":"05:01.440 ","End":"05:09.810","Text":"This just copying it from here and the n here is 2k minus 1."},{"Start":"05:09.810 ","End":"05:13.410","Text":"Now, minus 1 to the 2k is 1 because this is even."},{"Start":"05:13.410 ","End":"05:19.275","Text":"We can divide both sides by 2 and get Pi over 4 here to get rid of the 2 here."},{"Start":"05:19.275 ","End":"05:25.800","Text":"Pi over 4 equals the sum of 1 over 2k minus 1,"},{"Start":"05:25.800 ","End":"05:29.760","Text":"and this which alternates plus, minus, plus, minus."},{"Start":"05:29.760 ","End":"05:30.885","Text":"When k is 1,"},{"Start":"05:30.885 ","End":"05:33.570","Text":"we get minus 1^1 plus 1, it\u0027s plus 1."},{"Start":"05:33.570 ","End":"05:36.585","Text":"It starts with a 1 and then alternate plus or minus."},{"Start":"05:36.585 ","End":"05:38.070","Text":"What the series is,"},{"Start":"05:38.070 ","End":"05:40.260","Text":"is just 1 minus 1/3,"},{"Start":"05:40.260 ","End":"05:42.825","Text":"because 2k minus 1 runs over all the odd numbers."},{"Start":"05:42.825 ","End":"05:46.410","Text":"1 over 1 minus 1 over 3 plus 1 over 5, etc."},{"Start":"05:46.410 ","End":"05:50.230","Text":"That concludes this exercise."}],"ID":28742},{"Watched":false,"Name":"Exercise 1 - Part a","Duration":"1m 46s","ChapterTopicVideoID":27561,"CourseChapterTopicPlaylistID":294454,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.211","Text":"In this exercise, f is a 2Pi periodic function,"},{"Start":"00:04.211 ","End":"00:11.070","Text":"such that f is defined this way from minus Pi to 0 and this way from 0 to Pi."},{"Start":"00:11.070 ","End":"00:17.115","Text":"So effectively we\u0027ve defined it from minus Pi to Pi,"},{"Start":"00:17.115 ","End":"00:20.025","Text":"from here to here and from here to here."},{"Start":"00:20.025 ","End":"00:23.640","Text":"Notice that at 0,"},{"Start":"00:23.640 ","End":"00:27.210","Text":"this is equal to 2 and this is equal to 2."},{"Start":"00:27.210 ","End":"00:29.970","Text":"So it\u0027s well-defined at 0."},{"Start":"00:29.970 ","End":"00:34.145","Text":"If we just sketch it from minus Pi to Pi, it looks like this,"},{"Start":"00:34.145 ","End":"00:35.910","Text":"2 plus 2x or 2,"},{"Start":"00:35.910 ","End":"00:39.120","Text":"and it happens to be continuous at 0,"},{"Start":"00:39.120 ","End":"00:41.715","Text":"just mentioning that no particular reason."},{"Start":"00:41.715 ","End":"00:45.270","Text":"Now, given that it\u0027s 2Pi periodic,"},{"Start":"00:45.270 ","End":"00:47.255","Text":"means that the period of 2Pi,"},{"Start":"00:47.255 ","End":"00:49.978","Text":"so every 2Pi it repeats itself,"},{"Start":"00:49.978 ","End":"00:55.800","Text":"so let\u0027s repeat it on the left and let\u0027s repeat it on the right."},{"Start":"00:55.800 ","End":"01:00.600","Text":"This gives us from minus 3Pi to 3Pi."},{"Start":"01:00.600 ","End":"01:04.280","Text":"Of course, it continues infinitely in either direction."},{"Start":"01:04.280 ","End":"01:06.470","Text":"But this shows what it looks like."},{"Start":"01:06.470 ","End":"01:10.250","Text":"Note, that it\u0027s not continuous at Pi,"},{"Start":"01:10.250 ","End":"01:14.260","Text":"that it\u0027s not continuous at minus Pi, and in fact,"},{"Start":"01:14.260 ","End":"01:17.110","Text":"it\u0027s not continuous at 3Pi,"},{"Start":"01:17.110 ","End":"01:20.380","Text":"and it\u0027s not continuous at minus 3Pi,"},{"Start":"01:20.380 ","End":"01:22.723","Text":"because going to jump here also."},{"Start":"01:22.723 ","End":"01:27.290","Text":"What it means is that if we use the Dirichlet\u0027s theorem,"},{"Start":"01:27.290 ","End":"01:30.875","Text":"then here, for example, where it\u0027s continuous,"},{"Start":"01:30.875 ","End":"01:35.795","Text":"the series equals the value of the function but at this point and this point,"},{"Start":"01:35.795 ","End":"01:39.710","Text":"it\u0027ll be the average of the value on the left and on the right."},{"Start":"01:39.710 ","End":"01:42.260","Text":"Anyway, this is what the graph looks like,"},{"Start":"01:42.260 ","End":"01:44.525","Text":"and so we\u0027ve answered part a."},{"Start":"01:44.525 ","End":"01:47.100","Text":"That\u0027s it for this clip."}],"ID":28743},{"Watched":false,"Name":"Exercise 1 - Part b","Duration":"9m 23s","ChapterTopicVideoID":27562,"CourseChapterTopicPlaylistID":294454,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.435","Text":"Now we come to Part B of the exercise,"},{"Start":"00:03.435 ","End":"00:11.040","Text":"and we have to find the real Fourier series of f. The formula for the series is this."},{"Start":"00:11.040 ","End":"00:18.225","Text":"This is a general form and the coefficients a_n and b_n are given by formulas."},{"Start":"00:18.225 ","End":"00:21.065","Text":"For example, a_n is given by this formula."},{"Start":"00:21.065 ","End":"00:23.045","Text":"Let\u0027s compute that first."},{"Start":"00:23.045 ","End":"00:25.030","Text":"After we do a_n,"},{"Start":"00:25.030 ","End":"00:28.020","Text":"we\u0027ll do a_0 separately because as you\u0027ll see,"},{"Start":"00:28.020 ","End":"00:31.290","Text":"this formula doesn\u0027t quite work when n=0."},{"Start":"00:31.290 ","End":"00:33.495","Text":"Anyways, continue with this case."},{"Start":"00:33.495 ","End":"00:36.810","Text":"We replace f by what it\u0027s equal to,"},{"Start":"00:36.810 ","End":"00:40.395","Text":"but its piecewise is defined from minus Pi to 0,"},{"Start":"00:40.395 ","End":"00:42.000","Text":"this way that\u0027s f,"},{"Start":"00:42.000 ","End":"00:45.365","Text":"and f is 2 from 0 to Pi."},{"Start":"00:45.365 ","End":"00:47.240","Text":"Now I\u0027m going to use a trick here."},{"Start":"00:47.240 ","End":"00:48.373","Text":"You don\u0027t have to use this trick,"},{"Start":"00:48.373 ","End":"00:49.550","Text":"it can be done otherwise."},{"Start":"00:49.550 ","End":"00:56.285","Text":"But I want to change these limits to be from 0 to Pi also and then we can combine."},{"Start":"00:56.285 ","End":"00:59.495","Text":"If we replace x by minus x,"},{"Start":"00:59.495 ","End":"01:04.700","Text":"this is cheating because you can\u0027t replace x with minus x."},{"Start":"01:04.700 ","End":"01:09.350","Text":"You have to replace x with minus t and at the end switch from t back to x."},{"Start":"01:09.350 ","End":"01:13.190","Text":"But mentally we can imagine replacing x with minus x."},{"Start":"01:13.190 ","End":"01:17.630","Text":"For example, the cosine won\u0027t change if you replace x by minus x,"},{"Start":"01:17.630 ","End":"01:19.805","Text":"but dx will come out negative,"},{"Start":"01:19.805 ","End":"01:24.529","Text":"and also the limits of integration will be reversed."},{"Start":"01:24.529 ","End":"01:26.930","Text":"It will be from Pi to 0,"},{"Start":"01:26.930 ","End":"01:30.920","Text":"it\u0027s negative from 0 to Pi."},{"Start":"01:30.920 ","End":"01:35.945","Text":"Anyway, I claim that what we get here is integral 0 to pi, the same thing."},{"Start":"01:35.945 ","End":"01:39.980","Text":"We don\u0027t need any pluses or minuses because like I said,"},{"Start":"01:39.980 ","End":"01:46.640","Text":"we get a minus from here and we get a minus from reversing 0 and Pi."},{"Start":"01:46.640 ","End":"01:51.775","Text":"We also put a minus here because x goes to minus x."},{"Start":"01:51.775 ","End":"01:55.700","Text":"This now can be combined because they\u0027re both 0 to Pi."},{"Start":"01:55.700 ","End":"01:59.660","Text":"This 2 here, and these 2 here can make 4."},{"Start":"01:59.660 ","End":"02:04.310","Text":"We have a single function to integrate from 0 to Pi,"},{"Start":"02:04.310 ","End":"02:06.635","Text":"and then we can break this up."},{"Start":"02:06.635 ","End":"02:13.265","Text":"This is 4 times the integral of cosine nx dx."},{"Start":"02:13.265 ","End":"02:20.620","Text":"Then also, we can in this part take the 2 over Pi out and get xcosine nx."},{"Start":"02:20.620 ","End":"02:22.340","Text":"We have 2 integrals."},{"Start":"02:22.340 ","End":"02:28.595","Text":"We have a cosine nx dx and we have an xcosine nx dx."},{"Start":"02:28.595 ","End":"02:33.290","Text":"One way of doing the definite integral is to compute the indefinite integral,"},{"Start":"02:33.290 ","End":"02:36.305","Text":"and then substitute the limits of integration."},{"Start":"02:36.305 ","End":"02:40.750","Text":"For cosine nx we get sine nx over n,"},{"Start":"02:40.750 ","End":"02:44.880","Text":"and xcosine nx will give us this."},{"Start":"02:44.880 ","End":"02:49.145","Text":"I\u0027ll do the computation at the end so as not to break the flow."},{"Start":"02:49.145 ","End":"02:52.280","Text":"Now we just have to substitute limits."},{"Start":"02:52.280 ","End":"02:56.910","Text":"Sine nx is 0, both at x=0 and x=Pi,"},{"Start":"02:56.910 ","End":"02:58.695","Text":"we can ignore that."},{"Start":"02:58.695 ","End":"03:08.835","Text":"Also xsine nx is also going to be 0 at x=0 and x=Pi."},{"Start":"03:08.835 ","End":"03:11.550","Text":"All we\u0027re left with is this,"},{"Start":"03:11.550 ","End":"03:17.060","Text":"nx=Pi and nx=0 and subtract them and of course,"},{"Start":"03:17.060 ","End":"03:19.145","Text":"there this constant here in front,"},{"Start":"03:19.145 ","End":"03:24.705","Text":"combine the denominator n^2 and cosine nPi is minus 1 to the n,"},{"Start":"03:24.705 ","End":"03:27.480","Text":"and cosine 0 is 1."},{"Start":"03:27.480 ","End":"03:29.100","Text":"This is what we get."},{"Start":"03:29.100 ","End":"03:31.490","Text":"Now this is equal to;"},{"Start":"03:31.490 ","End":"03:33.080","Text":"if n is odd,"},{"Start":"03:33.080 ","End":"03:35.910","Text":"minus 1 minus 1 is minus 2,"},{"Start":"03:35.910 ","End":"03:37.140","Text":"and if n is even,"},{"Start":"03:37.140 ","End":"03:39.480","Text":"1 minus 1 is 0."},{"Start":"03:39.480 ","End":"03:42.375","Text":"We can write this out as follows."},{"Start":"03:42.375 ","End":"03:44.360","Text":"When n is odd, in other words,"},{"Start":"03:44.360 ","End":"03:46.945","Text":"when n is 2k minus 1."},{"Start":"03:46.945 ","End":"03:54.165","Text":"When we multiply out we get 4 over Pi^2 times n^2."},{"Start":"03:54.165 ","End":"03:58.050","Text":"But n^2 is 2k minus 1^2 when n is odd,"},{"Start":"03:58.050 ","End":"04:00.315","Text":"n is 2k minus 1."},{"Start":"04:00.315 ","End":"04:02.565","Text":"When n is even,"},{"Start":"04:02.565 ","End":"04:05.535","Text":"n equals 2k, we just get 0."},{"Start":"04:05.535 ","End":"04:09.720","Text":"That\u0027s a formula for a_n, odds and evens."},{"Start":"04:09.720 ","End":"04:13.620","Text":"Now this integral of cosine nx won\u0027t"},{"Start":"04:13.620 ","End":"04:18.250","Text":"work when n equal 0 because we\u0027re dividing by n everywhere."},{"Start":"04:18.250 ","End":"04:24.630","Text":"We\u0027ll have to do the case n equal 0 separately and let\u0027s go back to it. Here it is."},{"Start":"04:24.630 ","End":"04:27.065","Text":"This is the formula for a_0,"},{"Start":"04:27.065 ","End":"04:31.368","Text":"which is just gotten from this by putting n equal"},{"Start":"04:31.368 ","End":"04:36.305","Text":"0 and noticing that cosine of 0 is 1, so this disappears."},{"Start":"04:36.305 ","End":"04:38.630","Text":"This is equal to,"},{"Start":"04:38.630 ","End":"04:40.865","Text":"replace f by what it\u0027s equal,"},{"Start":"04:40.865 ","End":"04:42.935","Text":"which is piecewise this,"},{"Start":"04:42.935 ","End":"04:47.730","Text":"from minus Pi to 0 and 2 from 0 to Pi."},{"Start":"04:49.100 ","End":"04:52.995","Text":"This integral is 2x plus x^2 over Pi."},{"Start":"04:52.995 ","End":"04:54.330","Text":"Here 2x."},{"Start":"04:54.330 ","End":"04:56.430","Text":"Just plug in values."},{"Start":"04:56.430 ","End":"05:00.875","Text":"In other words it\u0027s just algebra."},{"Start":"05:00.875 ","End":"05:04.310","Text":"In the end we get that a_0=3."},{"Start":"05:04.310 ","End":"05:10.670","Text":"The last step we have 2Pi plus 2Pi is 4Pi minus Pi is 3Pi over Pi is 3."},{"Start":"05:10.670 ","End":"05:16.320","Text":"All together we have now a_n in general,"},{"Start":"05:16.320 ","End":"05:19.430","Text":"excluding 0 and here a_0 separately."},{"Start":"05:19.430 ","End":"05:22.425","Text":"What we\u0027re still missing is b_n."},{"Start":"05:22.425 ","End":"05:27.065","Text":"b_n is given by this formula with the sine nx,"},{"Start":"05:27.065 ","End":"05:30.320","Text":"and f is defined piecewise."},{"Start":"05:30.320 ","End":"05:38.890","Text":"From minus Pi to 0 it\u0027s defined this way and from 0 to Pi is defined as 2."},{"Start":"05:39.040 ","End":"05:42.530","Text":"I\u0027m going to use the same trick I did before to"},{"Start":"05:42.530 ","End":"05:45.800","Text":"make both of these integrals from 0 to Pi."},{"Start":"05:45.800 ","End":"05:49.805","Text":"If we replace x by minus x,"},{"Start":"05:49.805 ","End":"05:52.115","Text":"this is going to become a minus."},{"Start":"05:52.115 ","End":"05:54.260","Text":"The sine is odd,"},{"Start":"05:54.260 ","End":"05:55.745","Text":"so we\u0027ll get a minus."},{"Start":"05:55.745 ","End":"05:58.570","Text":"The dx will be minus."},{"Start":"05:58.570 ","End":"06:01.940","Text":"Also switching the integral,"},{"Start":"06:01.940 ","End":"06:04.430","Text":"the upper and lower limits, will also give a minus."},{"Start":"06:04.430 ","End":"06:05.990","Text":"It\u0027ll be 3 minuses,"},{"Start":"06:05.990 ","End":"06:08.345","Text":"which will make it a minus,"},{"Start":"06:08.345 ","End":"06:11.765","Text":"and this plus becomes a minus."},{"Start":"06:11.765 ","End":"06:17.225","Text":"Now, I\u0027ve marked these in red because they\u0027re going to cancel each other out."},{"Start":"06:17.225 ","End":"06:21.460","Text":"I have minus 2sine nx and plus 2sine nx."},{"Start":"06:21.460 ","End":"06:25.615","Text":"We can cancel now that the integrals are over the same interval."},{"Start":"06:25.615 ","End":"06:29.805","Text":"What we\u0027re left with is the following, let\u0027s see this."},{"Start":"06:29.805 ","End":"06:31.840","Text":"As I said, the 2 cancels."},{"Start":"06:31.840 ","End":"06:35.150","Text":"Minus with minus gives a plus,"},{"Start":"06:35.150 ","End":"06:36.905","Text":"so the minus disappears."},{"Start":"06:36.905 ","End":"06:41.280","Text":"This 2 goes in front here."},{"Start":"06:41.280 ","End":"06:50.650","Text":"This Pi with this Pi gives Pi^2 and we\u0027re just left with xsine nx,"},{"Start":"06:50.650 ","End":"06:51.935","Text":"the integral of that."},{"Start":"06:51.935 ","End":"06:53.570","Text":"Now this is a known integral,"},{"Start":"06:53.570 ","End":"06:55.430","Text":"but I\u0027m also going to compute it at the end."},{"Start":"06:55.430 ","End":"06:57.140","Text":"I\u0027ll just give you the answer now."},{"Start":"06:57.140 ","End":"07:01.220","Text":"The indefinite integral of xsinenx is the following."},{"Start":"07:01.220 ","End":"07:04.985","Text":"Now I would substitute 0 to Pi."},{"Start":"07:04.985 ","End":"07:07.010","Text":"Now there are 4 substitutions,"},{"Start":"07:07.010 ","End":"07:09.215","Text":"we have to substitute Pi here and here,"},{"Start":"07:09.215 ","End":"07:11.300","Text":"and 0 here and here."},{"Start":"07:11.300 ","End":"07:13.220","Text":"But only 1 of them is non-zero."},{"Start":"07:13.220 ","End":"07:20.670","Text":"Put Pi here and we get minus Pi cosine nPi over n,"},{"Start":"07:20.670 ","End":"07:27.645","Text":"but cosine nPi is minus 1 to the n. The Pi with the Pi^2 gives us just Pi,"},{"Start":"07:27.645 ","End":"07:29.865","Text":"there\u0027s an n in the denominator,"},{"Start":"07:29.865 ","End":"07:32.280","Text":"the minus in front."},{"Start":"07:32.280 ","End":"07:35.580","Text":"Here\u0027s the minus 1 to the n and here\u0027s the 2."},{"Start":"07:35.580 ","End":"07:39.750","Text":"This is what we have for b_n."},{"Start":"07:39.750 ","End":"07:43.755","Text":"Let\u0027s remember what we had for a_n and for a_0."},{"Start":"07:43.755 ","End":"07:46.865","Text":"Now we want to put these pieces together."},{"Start":"07:46.865 ","End":"07:49.810","Text":"We\u0027re going to put them in this formula,"},{"Start":"07:49.810 ","End":"07:53.040","Text":"a_0 is 3,"},{"Start":"07:53.040 ","End":"07:55.400","Text":"a_n is this,"},{"Start":"07:55.400 ","End":"07:58.805","Text":"we only take n equal 2k minus 1."},{"Start":"07:58.805 ","End":"08:00.975","Text":"We get the sum."},{"Start":"08:00.975 ","End":"08:04.290","Text":"K goes from 1 to infinity, and instead of n,"},{"Start":"08:04.290 ","End":"08:08.400","Text":"2k minus 1 here and here,"},{"Start":"08:08.400 ","End":"08:12.640","Text":"and b_n goes here."},{"Start":"08:12.770 ","End":"08:17.570","Text":"This is the Fourier series for f. I"},{"Start":"08:17.570 ","End":"08:22.420","Text":"just owe you the 2 integrals of xsine nx and xcosine nx."},{"Start":"08:22.420 ","End":"08:24.840","Text":"Let\u0027s start with xcosine nx,"},{"Start":"08:24.840 ","End":"08:26.445","Text":"we\u0027ll do it by parts."},{"Start":"08:26.445 ","End":"08:28.650","Text":"We\u0027ll let this be f and this be g\u0027,"},{"Start":"08:28.650 ","End":"08:36.740","Text":"so g is sine nx over n times x minus the integral of"},{"Start":"08:36.740 ","End":"08:45.440","Text":"f\u0027 g. This integral comes out to be minus cosine nx over n,"},{"Start":"08:45.440 ","End":"08:50.980","Text":"but the n with the n gives us n^2 and the minus with the minus makes it plus."},{"Start":"08:50.980 ","End":"08:55.940","Text":"This is the integral of that and that\u0027s what we used earlier."},{"Start":"08:55.940 ","End":"08:59.195","Text":"For this one, it\u0027s very similar."},{"Start":"08:59.195 ","End":"09:08.725","Text":"Again, we do it by parts and this time g is minus cosine nx over n, here and here."},{"Start":"09:08.725 ","End":"09:11.945","Text":"This integral comes out to be,"},{"Start":"09:11.945 ","End":"09:17.015","Text":"minus and minus is plus and the integral of cosine is sine."},{"Start":"09:17.015 ","End":"09:20.640","Text":"We get this and that\u0027s what we had earlier."},{"Start":"09:20.640 ","End":"09:24.000","Text":"That concludes part b."}],"ID":28744},{"Watched":false,"Name":"Exercise 1 - Part c","Duration":"4m 10s","ChapterTopicVideoID":27563,"CourseChapterTopicPlaylistID":294454,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.410","Text":"Now we come to part c where we have to prove that this series adds to Pi^2 over 8."},{"Start":"00:07.410 ","End":"00:10.830","Text":"I\u0027ll show you what this series is if we spell it out a bit."},{"Start":"00:10.830 ","End":"00:15.600","Text":"It\u0027s 1 over an odd number squared and we sum it for all the odd numbers,"},{"Start":"00:15.600 ","End":"00:17.430","Text":"so we have 1 over 1^2,"},{"Start":"00:17.430 ","End":"00:20.145","Text":"1 over 3^2, 1 over 5^2, 1 over 7^2."},{"Start":"00:20.145 ","End":"00:22.260","Text":"We add all these up to infinity,"},{"Start":"00:22.260 ","End":"00:25.590","Text":"we should get Pi^2 over 8."},{"Start":"00:25.590 ","End":"00:29.444","Text":"Numerically this is what it\u0027s equal to approximately."},{"Start":"00:29.444 ","End":"00:35.790","Text":"In part b, we found the Fourier series expansion for this f(x),"},{"Start":"00:35.790 ","End":"00:38.100","Text":"the real Fourier series."},{"Start":"00:38.100 ","End":"00:44.355","Text":"Now we want to apply the Dirichlet\u0027s theorem."},{"Start":"00:44.355 ","End":"00:47.368","Text":"The question is which point to substitute."},{"Start":"00:47.368 ","End":"00:52.010","Text":"I should say Parseval\u0027s theorem wouldn\u0027t work here in case you are thinking of trying it."},{"Start":"00:52.010 ","End":"00:53.165","Text":"You might think, aha,"},{"Start":"00:53.165 ","End":"00:55.220","Text":"we have 2n minus 1^2 here,"},{"Start":"00:55.220 ","End":"00:57.440","Text":"2k minus 1^2 here."},{"Start":"00:57.440 ","End":"01:01.550","Text":"Looks similar. But in Parseval\u0027s theorem,"},{"Start":"01:01.550 ","End":"01:03.590","Text":"we take the coefficients and we square them,"},{"Start":"01:03.590 ","End":"01:06.770","Text":"so we\u0027d get something 2k minus 1^4th,"},{"Start":"01:06.770 ","End":"01:08.075","Text":"so that\u0027s no good."},{"Start":"01:08.075 ","End":"01:10.010","Text":"Dirichlet\u0027s theorem is what we\u0027ll use."},{"Start":"01:10.010 ","End":"01:11.870","Text":"The question is which point to substitute?"},{"Start":"01:11.870 ","End":"01:15.035","Text":"Typically, it\u0027s one of the endpoints,"},{"Start":"01:15.035 ","End":"01:16.975","Text":"so minus Pi, 0, Pi,"},{"Start":"01:16.975 ","End":"01:20.555","Text":"or the center of the interval which is 0."},{"Start":"01:20.555 ","End":"01:24.500","Text":"If it\u0027s continuous at the point which will be true for 0,"},{"Start":"01:24.500 ","End":"01:27.035","Text":"we can just substitute it and get equality."},{"Start":"01:27.035 ","End":"01:31.670","Text":"But if we substitute one of the endpoints where it has a jump discontinuity,"},{"Start":"01:31.670 ","End":"01:35.185","Text":"then we have to take the average of the left limit and right limit."},{"Start":"01:35.185 ","End":"01:37.280","Text":"Anyway, in our case,"},{"Start":"01:37.280 ","End":"01:40.610","Text":"we\u0027ll take x=0 and at 0,"},{"Start":"01:40.610 ","End":"01:42.462","Text":"it\u0027s actually continuous,"},{"Start":"01:42.462 ","End":"01:48.350","Text":"so we can replace the Tilde with an equal sign if we plug in x=0."},{"Start":"01:48.350 ","End":"01:52.160","Text":"If we do that, what we get is, let\u0027s see,"},{"Start":"01:52.160 ","End":"01:54.810","Text":"x here is 0,"},{"Start":"01:54.810 ","End":"01:57.630","Text":"x here is 0."},{"Start":"01:57.630 ","End":"02:00.880","Text":"F of 0 is 2,"},{"Start":"02:00.880 ","End":"02:03.305","Text":"doesn\u0027t matter which formula you take it from,"},{"Start":"02:03.305 ","End":"02:08.090","Text":"constant 2 or 2 plus 2x over Pi when x is 0, so it\u0027s 2."},{"Start":"02:08.090 ","End":"02:10.450","Text":"On the right, what do we get?"},{"Start":"02:10.450 ","End":"02:14.014","Text":"Well, all these signs come out to be 0."},{"Start":"02:14.014 ","End":"02:15.470","Text":"I mean, sine of n,"},{"Start":"02:15.470 ","End":"02:16.880","Text":"0 is sine of 0 is 0,"},{"Start":"02:16.880 ","End":"02:19.465","Text":"so this whole series is 0."},{"Start":"02:19.465 ","End":"02:23.705","Text":"The cosine of something times 0 is cosine of 0,"},{"Start":"02:23.705 ","End":"02:25.070","Text":"which is 1,"},{"Start":"02:25.070 ","End":"02:26.705","Text":"so this just disappears."},{"Start":"02:26.705 ","End":"02:35.045","Text":"We have 4 over Pi^2 2k minus 1^2 and the sum of that from k=1 to infinity."},{"Start":"02:35.045 ","End":"02:37.865","Text":"Now just a bit of algebra, some manipulation,"},{"Start":"02:37.865 ","End":"02:41.000","Text":"2 minus 3 over 2 is a half,"},{"Start":"02:41.000 ","End":"02:43.910","Text":"and switch left and right-hand sides."},{"Start":"02:43.910 ","End":"02:46.985","Text":"Now I\u0027ve marked the 4 and the Pi^2."},{"Start":"02:46.985 ","End":"02:49.340","Text":"You want to bring those to the other side,"},{"Start":"02:49.340 ","End":"02:52.625","Text":"so we just have 1 over 2k minus 1^2."},{"Start":"02:52.625 ","End":"02:55.190","Text":"It\u0027s a half times Pi^2 over 4."},{"Start":"02:55.190 ","End":"02:57.950","Text":"That comes out to be Pi^2 over 8."},{"Start":"02:57.950 ","End":"03:00.680","Text":"Basically, this is what we have to prove."},{"Start":"03:00.680 ","End":"03:03.725","Text":"I could say we\u0027re done but hang on a moment,"},{"Start":"03:03.725 ","End":"03:06.320","Text":"just as a bonus or an optional extra,"},{"Start":"03:06.320 ","End":"03:08.840","Text":"I\u0027ll show you another way of computing it"},{"Start":"03:08.840 ","End":"03:11.510","Text":"from a known sum which we did in a different exercise."},{"Start":"03:11.510 ","End":"03:13.085","Text":"Or you might know it that the sum of"},{"Start":"03:13.085 ","End":"03:18.215","Text":"all the consecutive square reciprocals is Pi^2 over 6,"},{"Start":"03:18.215 ","End":"03:19.580","Text":"not Pi^2 over 8."},{"Start":"03:19.580 ","End":"03:22.235","Text":"We can use this to get our sum."},{"Start":"03:22.235 ","End":"03:25.774","Text":"If we take 1 over the odd numbers squared,"},{"Start":"03:25.774 ","End":"03:32.155","Text":"we can consider it as all the one over something squared minus 1 over evens squared."},{"Start":"03:32.155 ","End":"03:38.660","Text":"Now the evens, we can take out 1 over 2^2 from each of these and we have 1 over 1^2,"},{"Start":"03:38.660 ","End":"03:40.985","Text":"1 over 2, so 1 over 3^2 after that."},{"Start":"03:40.985 ","End":"03:43.070","Text":"Look at this and look at this."},{"Start":"03:43.070 ","End":"03:46.385","Text":"This is the original Pi^2 over 6."},{"Start":"03:46.385 ","End":"03:50.815","Text":"Anyway, it says it\u0027s 1 minus 1 over 2^2 times this."},{"Start":"03:50.815 ","End":"03:53.925","Text":"1 minus 1 over 2^2 is 3/4,"},{"Start":"03:53.925 ","End":"03:55.080","Text":"and this we already said,"},{"Start":"03:55.080 ","End":"03:56.880","Text":"is Pi^2 over 6."},{"Start":"03:56.880 ","End":"04:01.420","Text":"3/4 of Pi^2 over 6 is Pi^2 over 8."},{"Start":"04:01.420 ","End":"04:03.915","Text":"6 times 3 is 24 over 4 is 8."},{"Start":"04:03.915 ","End":"04:08.025","Text":"Pi^2 over 8, which is what we had just now anyway,"},{"Start":"04:08.025 ","End":"04:11.290","Text":"so that\u0027s good, and we are done."}],"ID":28745},{"Watched":false,"Name":"Exercise 2 - Part a","Duration":"3m 42s","ChapterTopicVideoID":27564,"CourseChapterTopicPlaylistID":294454,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.640","Text":"In this exercise, f(x) is x^2 on the interval minus Pi to Pi,"},{"Start":"00:05.640 ","End":"00:10.560","Text":"and extended periodically outside it belongs to the space of"},{"Start":"00:10.560 ","End":"00:16.470","Text":"piecewise continuous functions on minus Pi to Pi with the L^2 norm."},{"Start":"00:16.470 ","End":"00:20.220","Text":"Our task is to compute the real Fourier series of f and"},{"Start":"00:20.220 ","End":"00:24.405","Text":"then use this to compute 3 different series."},{"Start":"00:24.405 ","End":"00:27.179","Text":"For the first one we\u0027ll use Parseval\u0027s identity"},{"Start":"00:27.179 ","End":"00:29.670","Text":"and for these 2 will use the Dirichlet\u0027s theorem."},{"Start":"00:29.670 ","End":"00:32.175","Text":"Anyway, let\u0027s get started with Part a."},{"Start":"00:32.175 ","End":"00:36.175","Text":"In general, this is the form for a real Fourier series."},{"Start":"00:36.175 ","End":"00:37.460","Text":"But in our case,"},{"Start":"00:37.460 ","End":"00:39.260","Text":"f is an even function,"},{"Start":"00:39.260 ","End":"00:46.000","Text":"x^2 is even so all the b_n\u0027s are 0 and we just have the a_n\u0027s."},{"Start":"00:46.000 ","End":"00:49.650","Text":"F is of the form like this."},{"Start":"00:49.650 ","End":"00:51.930","Text":"We have a formula for a_n,"},{"Start":"00:51.930 ","End":"00:54.285","Text":"which is this into grow."},{"Start":"00:54.285 ","End":"00:56.704","Text":"Because f is even,"},{"Start":"00:56.704 ","End":"01:03.890","Text":"we also can rewrite this as twice the integral from 0 to Pi of f(x) cosine nxdx."},{"Start":"01:03.890 ","End":"01:08.040","Text":"Let\u0027s start computing a_n is this."},{"Start":"01:08.040 ","End":"01:13.910","Text":"I\u0027m going to use the formula for integration by parts, IBP for short."},{"Start":"01:13.910 ","End":"01:18.380","Text":"We\u0027ll take this as f. This is g\u0027."},{"Start":"01:18.380 ","End":"01:27.050","Text":"Our f\u0027 is the derivative of x^2 and g is the anti-derivative the integral of cosine nx,"},{"Start":"01:27.050 ","End":"01:32.930","Text":"which is sine x over n. If we plug in 0 and Pi,"},{"Start":"01:32.930 ","End":"01:35.120","Text":"then sine nx is 0."},{"Start":"01:35.120 ","End":"01:36.380","Text":"That\u0027s why I wrote this in gray."},{"Start":"01:36.380 ","End":"01:38.465","Text":"All this part is 0."},{"Start":"01:38.465 ","End":"01:42.490","Text":"Now we have this integral to compute."},{"Start":"01:42.490 ","End":"01:51.775","Text":"Now we can bring the minus and the 2 and the n in front minus 4 over nPi."},{"Start":"01:51.775 ","End":"01:55.795","Text":"Now we have the integral of x sine nx to compute,"},{"Start":"01:55.795 ","End":"01:58.670","Text":"I\u0027m going to use integration by parts again."},{"Start":"01:58.670 ","End":"02:02.780","Text":"This will be f, this will be g\u0027 in this round."},{"Start":"02:02.780 ","End":"02:07.070","Text":"We get g is minus cosine nx over n,"},{"Start":"02:07.070 ","End":"02:09.310","Text":"and f\u0027 is 1."},{"Start":"02:09.310 ","End":"02:12.570","Text":"In this part if we let x equals 0, we get 0."},{"Start":"02:12.570 ","End":"02:15.230","Text":"We just have the case where x is Pi,"},{"Start":"02:15.230 ","End":"02:19.670","Text":"and we get Pi minus cosine nPi over n. Here,"},{"Start":"02:19.670 ","End":"02:26.240","Text":"the integral of cosine nx is sine nx over n. I guess it should be n^2, sorry."},{"Start":"02:26.240 ","End":"02:28.370","Text":"Anyway, this comes out to be 0."},{"Start":"02:28.370 ","End":"02:29.900","Text":"The second part is 0,"},{"Start":"02:29.900 ","End":"02:32.420","Text":"and Pi sine nx is 0."},{"Start":"02:32.420 ","End":"02:35.195","Text":"Here, cosine nx,"},{"Start":"02:35.195 ","End":"02:37.490","Text":"which is cosine nPi is minus 1^n."},{"Start":"02:37.490 ","End":"02:42.825","Text":"The Pi with the Pi cancels."},{"Start":"02:42.825 ","End":"02:44.655","Text":"f minus 1 to the n,"},{"Start":"02:44.655 ","End":"02:46.815","Text":"n with n is n^2,"},{"Start":"02:46.815 ","End":"02:49.260","Text":"and minus with the minus gives a plus."},{"Start":"02:49.260 ","End":"02:52.275","Text":"We have 4 minus 1^n over n^2."},{"Start":"02:52.275 ","End":"02:56.120","Text":"All this is true when n is not equal to"},{"Start":"02:56.120 ","End":"02:59.990","Text":"0 because this integral wouldn\u0027t work with n equals 0."},{"Start":"02:59.990 ","End":"03:01.880","Text":"We do the 0 case separately."},{"Start":"03:01.880 ","End":"03:04.075","Text":"Let\u0027s do the 0 case."},{"Start":"03:04.075 ","End":"03:09.050","Text":"a_naught, drop the cosine of because when n is 0,"},{"Start":"03:09.050 ","End":"03:10.640","Text":"cosine of 0 is 1,"},{"Start":"03:10.640 ","End":"03:12.385","Text":"so we have this."},{"Start":"03:12.385 ","End":"03:15.390","Text":"This is x^3 over 3,"},{"Start":"03:15.390 ","End":"03:16.500","Text":"from 0 to Pi,"},{"Start":"03:16.500 ","End":"03:18.060","Text":"Pi^3 over 3,"},{"Start":"03:18.060 ","End":"03:21.840","Text":"and it simplifies just to 2Pi^2 over 3,"},{"Start":"03:21.840 ","End":"03:23.745","Text":"one of the Pi\u0027s cancels."},{"Start":"03:23.745 ","End":"03:27.600","Text":"Now we have a_naught and a_n."},{"Start":"03:27.600 ","End":"03:29.925","Text":"Now we want to combine them."},{"Start":"03:29.925 ","End":"03:33.390","Text":"In this formula, a_naught is 2 Pi^2 over 3,"},{"Start":"03:33.390 ","End":"03:36.480","Text":"so a_naught over 2 is just Pi^2 over 3,"},{"Start":"03:36.480 ","End":"03:39.270","Text":"and here we have the a_n."},{"Start":"03:39.270 ","End":"03:43.060","Text":"This is the answer for Part a."}],"ID":28746},{"Watched":false,"Name":"Exercise 2 - Part b","Duration":"1m 45s","ChapterTopicVideoID":27555,"CourseChapterTopicPlaylistID":294454,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.935","Text":"Now, we come to pop B,"},{"Start":"00:01.935 ","End":"00:06.510","Text":"which is to compute the sum of the series 1 over n^4."},{"Start":"00:06.510 ","End":"00:09.869","Text":"The result of part a was this."},{"Start":"00:09.869 ","End":"00:15.885","Text":"Now naught that we have here n^2 in the coefficients and here we have n^4."},{"Start":"00:15.885 ","End":"00:20.400","Text":"That\u0027s 1 indication you might want to use Parseval\u0027s identity."},{"Start":"00:20.400 ","End":"00:25.410","Text":"In general, if f has a Fourier series as follows,"},{"Start":"00:25.410 ","End":"00:30.885","Text":"then Parseval\u0027s identity is this will apply this in our case,"},{"Start":"00:30.885 ","End":"00:35.550","Text":"the b_n is 0, a_n is what\u0027s written here."},{"Start":"00:35.550 ","End":"00:38.130","Text":"And a_naught is this part."},{"Start":"00:38.130 ","End":"00:41.380","Text":"This is a_naught over 2 actually."},{"Start":"00:42.230 ","End":"00:44.340","Text":"a_naught is this."},{"Start":"00:44.340 ","End":"00:45.660","Text":"So we have a half v naught,"},{"Start":"00:45.660 ","End":"00:50.210","Text":"and this is the sum of an cosine nx."},{"Start":"00:50.210 ","End":"00:54.220","Text":"Parseval\u0027s identity here gives us that 1 of the Pi,"},{"Start":"00:54.220 ","End":"00:58.920","Text":"the integral of x^2 squared is a_naught squared over 2."},{"Start":"00:58.920 ","End":"01:04.550","Text":"This is a_naught squared over 2 and the sum of b_n^2."},{"Start":"01:04.550 ","End":"01:07.870","Text":"And here we get the integral of just x to the fourth."},{"Start":"01:07.870 ","End":"01:10.040","Text":"We don\u0027t need the absolute value."},{"Start":"01:10.040 ","End":"01:13.440","Text":"We get 4 over 9,"},{"Start":"01:13.440 ","End":"01:17.750","Text":"but times a half is 2 over 9 Pi^4."},{"Start":"01:17.750 ","End":"01:21.130","Text":"And here we get what we want with a coefficient,"},{"Start":"01:21.130 ","End":"01:25.080","Text":"but we have the 1 over n^4 and you have 4^2 is 16."},{"Start":"01:25.080 ","End":"01:30.215","Text":"But already we see that we can extract this series with a bit of algebra,"},{"Start":"01:30.215 ","End":"01:32.780","Text":"bring this to the left-hand side, this to the right,"},{"Start":"01:32.780 ","End":"01:37.660","Text":"and then divide by 16, cancel a bit."},{"Start":"01:37.760 ","End":"01:43.100","Text":"We end up with pi to the fourth over 90 as required."},{"Start":"01:43.100 ","End":"01:46.020","Text":"And so we are done."}],"ID":28747},{"Watched":false,"Name":"Exercise 2 - Part c","Duration":"1m 21s","ChapterTopicVideoID":27556,"CourseChapterTopicPlaylistID":294454,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"Now we come to part c of this exercise."},{"Start":"00:03.780 ","End":"00:07.920","Text":"The result of part a is this."},{"Start":"00:07.920 ","End":"00:12.600","Text":"We\u0027re going to use this and the Dirichlet\u0027s theorem to compute the sum"},{"Start":"00:12.600 ","End":"00:17.505","Text":"of the alternating series minus 1^n over n^2."},{"Start":"00:17.505 ","End":"00:23.760","Text":"Note that this here looks like this part here."},{"Start":"00:23.760 ","End":"00:26.145","Text":"If we let x=0,"},{"Start":"00:26.145 ","End":"00:28.095","Text":"cosine nx is 1,"},{"Start":"00:28.095 ","End":"00:30.675","Text":"we get very close to what we want."},{"Start":"00:30.675 ","End":"00:35.370","Text":"Take this and substitute x=0,"},{"Start":"00:35.370 ","End":"00:37.665","Text":"which we can do by Dirichlet."},{"Start":"00:37.665 ","End":"00:40.800","Text":"This, by the way, is what the function looks like,"},{"Start":"00:40.800 ","End":"00:43.820","Text":"it\u0027s x^2, but repeated periodically."},{"Start":"00:43.820 ","End":"00:45.605","Text":"We take the part from minus Pi to Pi."},{"Start":"00:45.605 ","End":"00:48.380","Text":"Anyway, at x=0,"},{"Start":"00:48.380 ","End":"00:51.125","Text":"we have a continuous function."},{"Start":"00:51.125 ","End":"00:52.520","Text":"By Dirichlet\u0027s theorem,"},{"Start":"00:52.520 ","End":"00:57.560","Text":"we can plug in x=0 and this too changes to an actual equality."},{"Start":"00:57.560 ","End":"01:04.360","Text":"We get 0^2 equals everything here just with x=0."},{"Start":"01:04.360 ","End":"01:07.470","Text":"Like we said, cosine 0 is 1."},{"Start":"01:07.470 ","End":"01:11.090","Text":"We just have to put this on the other side and"},{"Start":"01:11.090 ","End":"01:15.230","Text":"divide by 4 and we get the series that we want."},{"Start":"01:15.230 ","End":"01:19.025","Text":"The answer is minus Pi^2 over 12."},{"Start":"01:19.025 ","End":"01:21.930","Text":"That concludes part c."}],"ID":28748},{"Watched":false,"Name":"Exercise 2 - Part d","Duration":"2m 4s","ChapterTopicVideoID":27557,"CourseChapterTopicPlaylistID":294454,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.205","Text":"Now we come to part d where we have to compute the sum of this series."},{"Start":"00:06.205 ","End":"00:10.330","Text":"It says here to use this result."},{"Start":"00:10.330 ","End":"00:16.600","Text":"So the result from part a is this Fourier series."},{"Start":"00:16.600 ","End":"00:19.510","Text":"Now, in the previous part c,"},{"Start":"00:19.510 ","End":"00:21.985","Text":"we substituted the value x=0."},{"Start":"00:21.985 ","End":"00:26.815","Text":"Here\u0027s a picture of it."},{"Start":"00:26.815 ","End":"00:29.725","Text":"From minus Pi to Pi, then it continues."},{"Start":"00:29.725 ","End":"00:31.210","Text":"Zero is continuous,"},{"Start":"00:31.210 ","End":"00:33.355","Text":"so it can substitute the value."},{"Start":"00:33.355 ","End":"00:37.645","Text":"This time, we\u0027re going to substitute an endpoint Pi,"},{"Start":"00:37.645 ","End":"00:40.090","Text":"and because it\u0027s an endpoint,"},{"Start":"00:40.090 ","End":"00:45.230","Text":"we have to take the average of the value of the function,"},{"Start":"00:45.230 ","End":"00:52.775","Text":"that plus and that minus Pi from the right and that plus Pi from the left."},{"Start":"00:52.775 ","End":"00:58.150","Text":"Although really we can see it\u0027s continuous at these points if we continue it."},{"Start":"00:58.150 ","End":"01:00.200","Text":"We could just substitute the value,"},{"Start":"01:00.200 ","End":"01:02.750","Text":"but let\u0027s do it this way."},{"Start":"01:02.750 ","End":"01:07.935","Text":"We have the average of f(minus Pi) and f(Pi)."},{"Start":"01:07.935 ","End":"01:13.170","Text":"The function is x^2,"},{"Start":"01:13.170 ","End":"01:17.000","Text":"so here it\u0027s minus Pi^2."},{"Start":"01:17.000 ","End":"01:18.740","Text":"Here it\u0027s pi squared,"},{"Start":"01:18.740 ","End":"01:23.850","Text":"but comes out the same thing if you add an average, just Pi^2."},{"Start":"01:23.850 ","End":"01:32.930","Text":"I suppose, like I said, we could have just substituted x=Pi and got the Pi^2 right away."},{"Start":"01:32.930 ","End":"01:36.360","Text":"Anyway, we\u0027re at this point now."},{"Start":"01:36.360 ","End":"01:39.095","Text":"Then we can rearrange."},{"Start":"01:39.095 ","End":"01:43.865","Text":"What we can do is we can keep this series here,"},{"Start":"01:43.865 ","End":"01:48.050","Text":"bring the Pi^2 over 3 over to the right,"},{"Start":"01:48.050 ","End":"01:50.000","Text":"and then divide by 4,"},{"Start":"01:50.000 ","End":"01:51.515","Text":"which is a quarter."},{"Start":"01:51.515 ","End":"01:55.160","Text":"What we get is Pi^2 over 6."},{"Start":"01:55.160 ","End":"01:59.180","Text":"We get 2 over 3 Pi^2 and the quarter of 2 over 3 is a 6th."},{"Start":"01:59.180 ","End":"02:04.350","Text":"That\u0027s the answer to part d, and we\u0027re done."}],"ID":28749},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"5m ","ChapterTopicVideoID":27558,"CourseChapterTopicPlaylistID":294454,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.620","Text":"In this exercise, f(x) is cosine of ax,"},{"Start":"00:04.620 ","End":"00:06.675","Text":"a is some parameter,"},{"Start":"00:06.675 ","End":"00:09.435","Text":"but it must not be a whole number."},{"Start":"00:09.435 ","End":"00:11.955","Text":"It just won\u0027t work if it\u0027s a whole number."},{"Start":"00:11.955 ","End":"00:17.490","Text":"We don\u0027t want it to be one of the cosine nx in the Fourier series."},{"Start":"00:17.490 ","End":"00:22.785","Text":"We have to show that this is the Fourier expansion for f(x)."},{"Start":"00:22.785 ","End":"00:27.180","Text":"Using this, we have to prove 2 identities."},{"Start":"00:27.180 ","End":"00:32.025","Text":"One is the formula for the cosecant of Pi a."},{"Start":"00:32.025 ","End":"00:33.375","Text":"Well, I won\u0027t read it out."},{"Start":"00:33.375 ","End":"00:36.695","Text":"and the other one is the cotangent of Pi a."},{"Start":"00:36.695 ","End":"00:41.975","Text":"Actually it comes out the same thing except for the minus 1^n here."},{"Start":"00:41.975 ","End":"00:43.340","Text":"Let\u0027s start with part a,"},{"Start":"00:43.340 ","End":"00:45.305","Text":"which is the Fourier series."},{"Start":"00:45.305 ","End":"00:48.020","Text":"Now f is an even function."},{"Start":"00:48.020 ","End":"00:50.030","Text":"It\u0027s the cosine of ax."},{"Start":"00:50.030 ","End":"00:51.505","Text":"So cosine is even."},{"Start":"00:51.505 ","End":"00:55.550","Text":"All the bands are 0 and the Fourier series just has as in it."},{"Start":"00:55.550 ","End":"00:59.315","Text":"It\u0027s a_naught over 2 plus the sum of the a_n cosine nx."},{"Start":"00:59.315 ","End":"01:02.075","Text":"We have to find what these coefficients are."},{"Start":"01:02.075 ","End":"01:04.475","Text":"We have the formula that a_n equals this,"},{"Start":"01:04.475 ","End":"01:06.920","Text":"which is also true if n is 0."},{"Start":"01:06.920 ","End":"01:08.870","Text":"Because of the evenness,"},{"Start":"01:08.870 ","End":"01:16.035","Text":"we can replace the integral from minus Pi to Pi by twice the integral just from 0 to Pi."},{"Start":"01:16.035 ","End":"01:18.690","Text":"In our case, f(x) is cosine ax."},{"Start":"01:18.690 ","End":"01:21.225","Text":"This is our expression for a_n."},{"Start":"01:21.225 ","End":"01:23.190","Text":"Let\u0027s see if we can work it out."},{"Start":"01:23.190 ","End":"01:27.020","Text":"Here we\u0027ll need a trigonometric formula for cosine"},{"Start":"01:27.020 ","End":"01:31.970","Text":"Alpha cosine Beta the product of cosines and it\u0027s equal to this."},{"Start":"01:31.970 ","End":"01:33.715","Text":"I\u0027m applying that here,"},{"Start":"01:33.715 ","End":"01:36.805","Text":"where Alpha is ax Beta is nx."},{"Start":"01:36.805 ","End":"01:39.655","Text":"We get, well, ax plus nx,"},{"Start":"01:39.655 ","End":"01:42.785","Text":"we can take x out the brackets and here also,"},{"Start":"01:42.785 ","End":"01:45.590","Text":"then we can do the integral."},{"Start":"01:45.590 ","End":"01:48.200","Text":"We know that the integral of cosine is"},{"Start":"01:48.200 ","End":"01:51.769","Text":"sine but adjusted because of the internal derivative,"},{"Start":"01:51.769 ","End":"01:55.070","Text":"which is a plus n, we have to divide by that."},{"Start":"01:55.070 ","End":"01:58.745","Text":"This is not 0 because we already said that"},{"Start":"01:58.745 ","End":"02:03.019","Text":"a is not a whole number and the same for the other term,"},{"Start":"02:03.019 ","End":"02:06.425","Text":"also, not 0 in the denominator."},{"Start":"02:06.425 ","End":"02:09.075","Text":"Then we can substitute."},{"Start":"02:09.075 ","End":"02:10.995","Text":"When x is 0,"},{"Start":"02:10.995 ","End":"02:14.300","Text":"both of these are 0 because we get sine 0,"},{"Start":"02:14.300 ","End":"02:16.070","Text":"sine 0, which is 0."},{"Start":"02:16.070 ","End":"02:18.040","Text":"So we just have to plug in the x=Pi."},{"Start":"02:18.040 ","End":"02:25.725","Text":"We replace this x by Pi and this x by Pi and now we have what an is."},{"Start":"02:25.725 ","End":"02:28.380","Text":"But let\u0027s develop this more."},{"Start":"02:28.380 ","End":"02:32.345","Text":"The sine of Pi a plus Pi n,"},{"Start":"02:32.345 ","End":"02:34.775","Text":"we can use another trig formula,"},{"Start":"02:34.775 ","End":"02:39.350","Text":"more familiar one for the sine of Alpha plus Beta plus or minus"},{"Start":"02:39.350 ","End":"02:43.960","Text":"and then we get sine over cosine Beta plus cosine Alpha sine Beta,"},{"Start":"02:43.960 ","End":"02:46.475","Text":"or minus if it\u0027s minus here."},{"Start":"02:46.475 ","End":"02:50.150","Text":"Yeah, so this is equal to this times this plus this times this."},{"Start":"02:50.150 ","End":"02:55.075","Text":"Now cosine of Pi n is minus 1^n and sine Pi n is 0,"},{"Start":"02:55.075 ","End":"03:00.990","Text":"so this comes out to be minus 1^n sine Pi a."},{"Start":"03:00.990 ","End":"03:04.035","Text":"The second numerator here,"},{"Start":"03:04.035 ","End":"03:05.555","Text":"is exactly the same."},{"Start":"03:05.555 ","End":"03:06.920","Text":"It\u0027s almost identical."},{"Start":"03:06.920 ","End":"03:09.140","Text":"Now we can put these 2 here."},{"Start":"03:09.140 ","End":"03:19.230","Text":"The sine of Pi plus an is equal to minus 1^n sine Pi a, which is this."},{"Start":"03:19.230 ","End":"03:21.214","Text":"The other one is the same."},{"Start":"03:21.214 ","End":"03:23.435","Text":"This one is also,"},{"Start":"03:23.435 ","End":"03:24.575","Text":"it\u0027s equal to this,"},{"Start":"03:24.575 ","End":"03:28.910","Text":"but these 2 are the same so we put that here,"},{"Start":"03:28.910 ","End":"03:30.350","Text":"just take it out the bracket,"},{"Start":"03:30.350 ","End":"03:32.330","Text":"leave 1 over a plus n,"},{"Start":"03:32.330 ","End":"03:34.505","Text":"1 over a minus n,"},{"Start":"03:34.505 ","End":"03:38.985","Text":"and the minus 1^n sine Pi a from here,"},{"Start":"03:38.985 ","End":"03:41.495","Text":"and 1 over Pi was already here."},{"Start":"03:41.495 ","End":"03:43.775","Text":"So this is a_n."},{"Start":"03:43.775 ","End":"03:49.010","Text":"Now this also works for n=0 because there\u0027s no problem if n is zeros,"},{"Start":"03:49.010 ","End":"03:51.470","Text":"no denominator 0 or anything."},{"Start":"03:51.470 ","End":"03:53.855","Text":"We just have to put in n=0."},{"Start":"03:53.855 ","End":"03:57.560","Text":"We\u0027d like to have a_0 separately because in the formula for"},{"Start":"03:57.560 ","End":"04:02.250","Text":"the Fourier series here we write a_naught separately."},{"Start":"04:02.250 ","End":"04:05.130","Text":"So this is a_naught."},{"Start":"04:05.130 ","End":"04:06.400","Text":"When n is naught,"},{"Start":"04:06.400 ","End":"04:10.325","Text":"this disappears, and this disappears,"},{"Start":"04:10.325 ","End":"04:13.825","Text":"and this disappears, and this disappears,"},{"Start":"04:13.825 ","End":"04:19.005","Text":"and we have twice sine Pi a over a."},{"Start":"04:19.005 ","End":"04:23.840","Text":"This is the 2 from it appearing twice and so this is what we have."},{"Start":"04:23.840 ","End":"04:25.895","Text":"We have a_n and we have a_naught."},{"Start":"04:25.895 ","End":"04:27.210","Text":"We didn\u0027t need b_n."},{"Start":"04:27.210 ","End":"04:28.935","Text":"Remember, we know the b_ns are all 0."},{"Start":"04:28.935 ","End":"04:32.280","Text":"Now we want to plug this and this in the formula."},{"Start":"04:32.280 ","End":"04:36.875","Text":"First, let\u0027s see what a over 2 is because we need it in this form."},{"Start":"04:36.875 ","End":"04:39.590","Text":"We just divide the 2 by the 2,"},{"Start":"04:39.590 ","End":"04:43.190","Text":"2 disappears, put the Pi together with the a."},{"Start":"04:43.190 ","End":"04:44.720","Text":"Now in this formula,"},{"Start":"04:44.720 ","End":"04:46.610","Text":"we have a_naught over 2,"},{"Start":"04:46.610 ","End":"04:53.885","Text":"which is sine Pi a over Pi a and we have a_n and just stick on a cosine nx to it."},{"Start":"04:53.885 ","End":"04:59.150","Text":"This is the Fourier series for f(x)."},{"Start":"04:59.150 ","End":"05:01.650","Text":"That concludes part a."}],"ID":28750},{"Watched":false,"Name":"Exercise 3 - Part b","Duration":"3m 34s","ChapterTopicVideoID":27559,"CourseChapterTopicPlaylistID":294454,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.415","Text":"Now we come to part b."},{"Start":"00:02.415 ","End":"00:07.470","Text":"In part a, we showed that the function cos(ax) can be"},{"Start":"00:07.470 ","End":"00:12.690","Text":"expanded as a Fourier series as follows and in part b,"},{"Start":"00:12.690 ","End":"00:15.090","Text":"we\u0027re going to use this to prove a couple of"},{"Start":"00:15.090 ","End":"00:20.190","Text":"identities about the cosecant and about the cotangent of Pi a."},{"Start":"00:20.190 ","End":"00:22.535","Text":"Let\u0027s start with the first one."},{"Start":"00:22.535 ","End":"00:26.675","Text":"We start with the Fourier series from part A, which is this."},{"Start":"00:26.675 ","End":"00:30.050","Text":"What we\u0027re going to do is substitute the value of x according"},{"Start":"00:30.050 ","End":"00:34.040","Text":"to Dirichlet\u0027s theorem and that should give us what we need."},{"Start":"00:34.040 ","End":"00:39.170","Text":"Now, this series here has a lot similar with this series here."},{"Start":"00:39.170 ","End":"00:41.125","Text":"We have the ( minus 1)^n,"},{"Start":"00:41.125 ","End":"00:42.265","Text":"we have the Pi,"},{"Start":"00:42.265 ","End":"00:44.990","Text":"we don\u0027t have sin(Pi a),"},{"Start":"00:44.990 ","End":"00:46.730","Text":"we have this here."},{"Start":"00:46.730 ","End":"00:52.220","Text":"But what we\u0027re going to do is if sin(Pi a) goes to the other side,"},{"Start":"00:52.220 ","End":"00:54.080","Text":"it will be cosec(Pi a)."},{"Start":"00:54.080 ","End":"00:57.800","Text":"Also, we have here cos(nx),"},{"Start":"00:57.800 ","End":"01:00.364","Text":"and here we have cos(ax),"},{"Start":"01:00.364 ","End":"01:02.310","Text":"but if x is 0,"},{"Start":"01:02.310 ","End":"01:04.600","Text":"both of these come out to be 1."},{"Start":"01:04.600 ","End":"01:06.060","Text":"So it\u0027s looking good."},{"Start":"01:06.060 ","End":"01:14.820","Text":"What we\u0027re going to do is let x=0 so that cos(ax) is 1. Let\u0027s see this."},{"Start":"01:14.820 ","End":"01:18.345","Text":"x=0. Cos(ax), which is cos(a0), is 1."},{"Start":"01:18.345 ","End":"01:21.165","Text":"Cos(nx) is cos(n0) is 1."},{"Start":"01:21.165 ","End":"01:24.750","Text":"Here we have the sin(Pi a) and the sin(Pi a)."},{"Start":"01:24.750 ","End":"01:29.983","Text":"We\u0027re going to bring this over to the left so we have 1 over sin(Pi a),"},{"Start":"01:29.983 ","End":"01:35.970","Text":"and everything that\u0027s left on the right is exactly what we need for this cosecant."},{"Start":"01:35.970 ","End":"01:40.250","Text":"1 over sin(Pi a) is cosec(Pi a) and the right-hand side"},{"Start":"01:40.250 ","End":"01:44.915","Text":"is exactly the series that we had to prove so that\u0027s the first one of 2."},{"Start":"01:44.915 ","End":"01:48.080","Text":"Now let\u0027s do the second one with the cotangent."},{"Start":"01:48.080 ","End":"01:53.300","Text":"We start where we were before with the cos(ax) having the following expansion,"},{"Start":"01:53.300 ","End":"01:54.980","Text":"we got that from here."},{"Start":"01:54.980 ","End":"01:57.170","Text":"Now we\u0027re going to substitute something else,"},{"Start":"01:57.170 ","End":"01:58.580","Text":"this time not 0."},{"Start":"01:58.580 ","End":"02:02.780","Text":"One of the things that points us is this minus ( minus 1)^n,"},{"Start":"02:02.780 ","End":"02:04.415","Text":"which is missing here."},{"Start":"02:04.415 ","End":"02:07.820","Text":"We know that cos(n Pi) is ( minus 1)^n,"},{"Start":"02:07.820 ","End":"02:09.665","Text":"so how about substituting Pi,"},{"Start":"02:09.665 ","End":"02:11.675","Text":"which is 1 of the endpoints."},{"Start":"02:11.675 ","End":"02:13.400","Text":"Now because it\u0027s an endpoint,"},{"Start":"02:13.400 ","End":"02:18.025","Text":"one of the parts of the Dirichlet theorem is that if we do that,"},{"Start":"02:18.025 ","End":"02:19.975","Text":"we substitute in f,"},{"Start":"02:19.975 ","End":"02:24.030","Text":"the average of the values of f at both endpoints,"},{"Start":"02:24.030 ","End":"02:28.625","Text":"at minus Pi from the right and at plus Pi from the left."},{"Start":"02:28.625 ","End":"02:35.450","Text":"What we get is f of minus Pi is cos(minus Pi a),"},{"Start":"02:35.450 ","End":"02:38.135","Text":"which is this part here."},{"Start":"02:38.135 ","End":"02:39.770","Text":"Then also on the right,"},{"Start":"02:39.770 ","End":"02:41.690","Text":"we have cos(Pi a),"},{"Start":"02:41.690 ","End":"02:43.460","Text":"which is this value here,"},{"Start":"02:43.460 ","End":"02:45.560","Text":"divided by 2 for the average,"},{"Start":"02:45.560 ","End":"02:48.740","Text":"and turns out we could have just substituted Pi a,"},{"Start":"02:48.740 ","End":"02:51.320","Text":"but that\u0027s not according to the theorem."},{"Start":"02:51.320 ","End":"02:55.580","Text":"It comes out to be cos(Pi a) here and here,"},{"Start":"02:55.580 ","End":"02:59.300","Text":"like I said, we get ( minus 1)^n because we get cos(Pi n)."},{"Start":"02:59.300 ","End":"03:04.240","Text":"Now, this (minus 1)^n will cancel with this (minus 1)^n."},{"Start":"03:04.240 ","End":"03:06.140","Text":"Here we\u0027ll have cos(Pi a)."},{"Start":"03:06.140 ","End":"03:11.795","Text":"When we divide by sin(Pi a) then we will get cotan(Pi a)."},{"Start":"03:11.795 ","End":"03:14.330","Text":"So cos(Pi a) over the sine,"},{"Start":"03:14.330 ","End":"03:17.633","Text":"that leaves just a 1 here and a 1 here."},{"Start":"03:17.633 ","End":"03:22.540","Text":"This (minus 1)^n with this (minus 1)^n"},{"Start":"03:22.540 ","End":"03:28.400","Text":"cancels and so we\u0027re just left with cotan(Pi a) equals this,"},{"Start":"03:28.400 ","End":"03:30.425","Text":"which is what we had to show."},{"Start":"03:30.425 ","End":"03:35.130","Text":"Yeah, there we got exactly this and so we are done."}],"ID":28751}],"Thumbnail":null,"ID":294454},{"Name":"Even and Odd Extensions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Even Fourier Extension","Duration":"5m 28s","ChapterTopicVideoID":27547,"CourseChapterTopicPlaylistID":294455,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.735","Text":"New topic, even and odd extensions."},{"Start":"00:03.735 ","End":"00:06.090","Text":"We\u0027ll start with even extensions."},{"Start":"00:06.090 ","End":"00:07.470","Text":"Here\u0027s the situation,"},{"Start":"00:07.470 ","End":"00:10.155","Text":"we\u0027re given a piecewise continuous function,"},{"Start":"00:10.155 ","End":"00:12.285","Text":"but not on minus Pi, Pi,"},{"Start":"00:12.285 ","End":"00:15.210","Text":"just on half of it from 0-Pi."},{"Start":"00:15.210 ","End":"00:18.405","Text":"How can we build a Fourier series for it?"},{"Start":"00:18.405 ","End":"00:22.698","Text":"One way to do it is to extend it to minus Pi,"},{"Start":"00:22.698 ","End":"00:24.600","Text":"Pi, and we\u0027ll talk about how."},{"Start":"00:24.600 ","End":"00:28.028","Text":"Then build the usual Fourier series on minus Pi,"},{"Start":"00:28.028 ","End":"00:32.765","Text":"Pi and then get our answer and just restricted to 0Pi."},{"Start":"00:32.765 ","End":"00:36.350","Text":"There are any number of ways to extend it in infinite number of"},{"Start":"00:36.350 ","End":"00:39.950","Text":"ways you could define it how you like from minus Pi-0."},{"Start":"00:39.950 ","End":"00:44.345","Text":"But there are two interesting ways in the context of Fourier series,"},{"Start":"00:44.345 ","End":"00:47.020","Text":"those are even and odd extensions."},{"Start":"00:47.020 ","End":"00:49.820","Text":"One way is to extend to an even function,"},{"Start":"00:49.820 ","End":"00:52.435","Text":"call it g as follows,"},{"Start":"00:52.435 ","End":"01:01.310","Text":"g is exactly the same as f on the interval from 0-Pi and from minus Pi-0,"},{"Start":"01:01.310 ","End":"01:04.345","Text":"we define g(x) as f(-x)."},{"Start":"01:04.345 ","End":"01:07.670","Text":"If this is the graph of f,"},{"Start":"01:07.670 ","End":"01:10.460","Text":"then the graph of g look like this,"},{"Start":"01:10.460 ","End":"01:12.080","Text":"like making a mirror image,"},{"Start":"01:12.080 ","End":"01:15.590","Text":"completing it to make it symmetric around the y-axis,"},{"Start":"01:15.590 ","End":"01:17.840","Text":"so it\u0027s an even function."},{"Start":"01:17.840 ","End":"01:21.605","Text":"Note that g, two things about it."},{"Start":"01:21.605 ","End":"01:24.680","Text":"First of all, it\u0027s an even function, and secondly,"},{"Start":"01:24.680 ","End":"01:28.220","Text":"it\u0027s the same as f on 0Pi,"},{"Start":"01:28.220 ","End":"01:31.255","Text":"and that\u0027s why we call it an even extension."},{"Start":"01:31.255 ","End":"01:33.090","Text":"That\u0027s the technical term,"},{"Start":"01:33.090 ","End":"01:37.655","Text":"g is an even extension of f. Now because g is even,"},{"Start":"01:37.655 ","End":"01:41.660","Text":"the Fourier series won\u0027t have any signs in it,"},{"Start":"01:41.660 ","End":"01:44.485","Text":"only the cosine and the constant term,"},{"Start":"01:44.485 ","End":"01:48.799","Text":"so that g will be of the following form or cosine."},{"Start":"01:48.799 ","End":"01:54.800","Text":"You could say the constant is a cosine also it\u0027s like cosine 0x is 1 anyway."},{"Start":"01:54.800 ","End":"01:57.650","Text":"Then we basically say that f is the same,"},{"Start":"01:57.650 ","End":"02:02.345","Text":"except that f(x) is restricted to the interval 0Pi."},{"Start":"02:02.345 ","End":"02:08.375","Text":"That\u0027s how we find the Fourier series for a function that\u0027s defined only from 0-Pi."},{"Start":"02:08.375 ","End":"02:14.420","Text":"This is also called the cosine series of f for obvious reasons."},{"Start":"02:14.420 ","End":"02:16.775","Text":"Now let\u0027s do an example."},{"Start":"02:16.775 ","End":"02:19.880","Text":"Our task is to find the cosine series for"},{"Start":"02:19.880 ","End":"02:26.655","Text":"the function f(x)=x on the interval from 0-Pi. That\u0027s part 1."},{"Start":"02:26.655 ","End":"02:31.190","Text":"To prove that for all x strictly between 0 and Pi,"},{"Start":"02:31.190 ","End":"02:34.630","Text":"we have the following equality."},{"Start":"02:34.630 ","End":"02:36.900","Text":"We follow the procedure above,"},{"Start":"02:36.900 ","End":"02:40.898","Text":"we define an even function g on all of minus Pi,"},{"Start":"02:40.898 ","End":"02:43.145","Text":"Pi using the following formula."},{"Start":"02:43.145 ","End":"02:47.180","Text":"Since f(x)=x here, this would be f(-x)."},{"Start":"02:47.180 ","End":"02:51.430","Text":"It just comes out to be the absolute value of x if you think about it."},{"Start":"02:51.430 ","End":"02:54.930","Text":"Next, we\u0027re going to express g as"},{"Start":"02:54.930 ","End":"02:58.880","Text":"a cosine series and we actually have a formula for the coefficient,"},{"Start":"02:58.880 ","End":"03:01.850","Text":"say n, and this is the formula."},{"Start":"03:01.850 ","End":"03:03.655","Text":"Now let\u0027s start computing."},{"Start":"03:03.655 ","End":"03:06.780","Text":"This is a_n because it\u0027s an even function,"},{"Start":"03:06.780 ","End":"03:10.590","Text":"we can put a 2 here and just take it from 0-Pi."},{"Start":"03:10.590 ","End":"03:16.995","Text":"But g(x)=x on the interval from 0-Pi."},{"Start":"03:16.995 ","End":"03:20.475","Text":"This x cosine nx will do by parts."},{"Start":"03:20.475 ","End":"03:23.235","Text":"This is f and g^1."},{"Start":"03:23.235 ","End":"03:25.935","Text":"This is going to equal f(g),"},{"Start":"03:25.935 ","End":"03:27.920","Text":"g is the integral of this,"},{"Start":"03:27.920 ","End":"03:32.835","Text":"which is sine nx minus the integral of f^1(g),"},{"Start":"03:32.835 ","End":"03:34.380","Text":"f^1 is 1,"},{"Start":"03:34.380 ","End":"03:37.020","Text":"g is same as here."},{"Start":"03:37.020 ","End":"03:43.635","Text":"This part is 0 because sine nPi is 0."},{"Start":"03:43.635 ","End":"03:48.680","Text":"This integral comes out to be the integral of minus sine is"},{"Start":"03:48.680 ","End":"03:53.555","Text":"cosine and we have to divide by another n and the 2/Pi is here."},{"Start":"03:53.555 ","End":"03:55.330","Text":"If we compute this,"},{"Start":"03:55.330 ","End":"03:57.285","Text":"this comes out to be,"},{"Start":"03:57.285 ","End":"04:01.165","Text":"we\u0027ve done this before, cosine nx is -1^n."},{"Start":"04:01.165 ","End":"04:06.470","Text":"This is what we have and we just separate the evens and the odds,"},{"Start":"04:06.470 ","End":"04:10.025","Text":"so if n is even,"},{"Start":"04:10.025 ","End":"04:13.980","Text":"then we get 1 minus 1 is 0."},{"Start":"04:14.150 ","End":"04:18.530","Text":"I forgot to say earlier that this passage from here to"},{"Start":"04:18.530 ","End":"04:22.370","Text":"here only works if n is not equal to 0 because we can\u0027t divide by n,"},{"Start":"04:22.370 ","End":"04:27.200","Text":"we\u0027re going to have to compute the a_0 coefficient separately after this."},{"Start":"04:27.200 ","End":"04:30.244","Text":"This is when n is even but not 0."},{"Start":"04:30.244 ","End":"04:33.155","Text":"What we get for a _ is 2/Pi,"},{"Start":"04:33.155 ","End":"04:35.830","Text":"the integral of x(dx),"},{"Start":"04:35.830 ","End":"04:41.725","Text":"because this part is when n is 0, is just x(dx)."},{"Start":"04:41.725 ","End":"04:49.090","Text":"This is equal to 2/Pi integral of x is x^2/2 substitute so we get 2/Pi,"},{"Start":"04:49.090 ","End":"04:51.420","Text":"Pi^2/2, which is Pi."},{"Start":"04:51.420 ","End":"04:57.780","Text":"Now we have all the coefficients and we can say that g(x) is equal to a_0/2,"},{"Start":"04:57.780 ","End":"05:03.335","Text":"which is Pi/2 plus Sigma of what\u0027s here,"},{"Start":"05:03.335 ","End":"05:06.200","Text":"times cosine and not n,"},{"Start":"05:06.200 ","End":"05:10.335","Text":"but 2k minus 1 times x."},{"Start":"05:10.335 ","End":"05:13.155","Text":"This is where x is in the interval of minus Pi-Pi."},{"Start":"05:13.155 ","End":"05:15.890","Text":"Now all we have to do is restrict it,"},{"Start":"05:15.890 ","End":"05:20.180","Text":"which is just declaring that x only belongs to the interval 0Pi,"},{"Start":"05:20.180 ","End":"05:23.245","Text":"but the same expression here."},{"Start":"05:23.245 ","End":"05:26.775","Text":"This is the Fourier series for f,"},{"Start":"05:26.775 ","End":"05:29.020","Text":"and we are done."}],"ID":28752},{"Watched":false,"Name":"Odd Fourier Extension","Duration":"4m 50s","ChapterTopicVideoID":27548,"CourseChapterTopicPlaylistID":294455,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"Continuing with even and odd extensions."},{"Start":"00:02.700 ","End":"00:04.710","Text":"We\u0027ve just talked about even extension,"},{"Start":"00:04.710 ","End":"00:07.680","Text":"so this time we\u0027ll talk about odd extensions."},{"Start":"00:07.680 ","End":"00:16.320","Text":"Quite similar, we extend f to an odd function, g on the interval from minus Pi to"},{"Start":"00:16.320 ","End":"00:25.775","Text":"Pi by letting g(x)=f(x) from 0 to Pi, but minus f(-x)."},{"Start":"00:25.775 ","End":"00:32.029","Text":"This time, there\u0027s a minus as opposed to the even case when x is between minus Pi and 0."},{"Start":"00:32.029 ","End":"00:35.615","Text":"If this is the function f on 0, Pi,"},{"Start":"00:35.615 ","End":"00:40.200","Text":"then the function g is just flip"},{"Start":"00:40.200 ","End":"00:45.785","Text":"this 180 degrees around and put it here and this is what we get roughly."},{"Start":"00:45.785 ","End":"00:50.345","Text":"G is an odd function. It\u0027s clear."},{"Start":"00:50.345 ","End":"00:53.660","Text":"Also it\u0027s the same as f on 0, Pi,"},{"Start":"00:53.660 ","End":"00:56.330","Text":"so it extends f and it\u0027s called the odd"},{"Start":"00:56.330 ","End":"01:00.230","Text":"extension of f. This time as opposed to the even case,"},{"Start":"01:00.230 ","End":"01:03.640","Text":"we\u0027ll only have sines and no cosines."},{"Start":"01:03.640 ","End":"01:07.665","Text":"We\u0027ll get g of the form the sum of b_n sine nx."},{"Start":"01:07.665 ","End":"01:10.895","Text":"After we found the b_n and got the series for g,"},{"Start":"01:10.895 ","End":"01:12.920","Text":"we just restrict it to 0, Pi."},{"Start":"01:12.920 ","End":"01:17.290","Text":"We just declare that this is valid on 0 to Pi."},{"Start":"01:17.290 ","End":"01:22.640","Text":"This is called the sine series of f. Let\u0027s do an example."},{"Start":"01:22.640 ","End":"01:29.090","Text":"We have to find the sine series for the function f(x)=1 on 0, Pi."},{"Start":"01:29.090 ","End":"01:33.550","Text":"Then we have to prove that A,"},{"Start":"01:33.550 ","End":"01:36.230","Text":"for all x strictly between 0 and Pi,"},{"Start":"01:36.230 ","End":"01:40.295","Text":"we have the following equality and B,"},{"Start":"01:40.295 ","End":"01:44.330","Text":"we have the following infinite series for Pi over 4."},{"Start":"01:44.330 ","End":"01:46.430","Text":"As the procedure says,"},{"Start":"01:46.430 ","End":"01:51.935","Text":"we define an odd function g and all of minus Pi, Pi as follows."},{"Start":"01:51.935 ","End":"01:58.635","Text":"Then we say that g(x) is represented by Fourier series of just sines."},{"Start":"01:58.635 ","End":"02:01.545","Text":"We have a formula for the b_n,"},{"Start":"02:01.545 ","End":"02:06.935","Text":"1 over Pi times the integral of g times sine nx."},{"Start":"02:06.935 ","End":"02:11.220","Text":"By the way, this is what g(x) looks like."},{"Start":"02:11.220 ","End":"02:12.855","Text":"It\u0027s 1 here,"},{"Start":"02:12.855 ","End":"02:15.195","Text":"but it\u0027s minus 1 here."},{"Start":"02:15.195 ","End":"02:18.380","Text":"Now let\u0027s start computing this integral."},{"Start":"02:18.380 ","End":"02:20.390","Text":"Because this is an even function,"},{"Start":"02:20.390 ","End":"02:22.640","Text":"it\u0027s an odd function times an odd function,"},{"Start":"02:22.640 ","End":"02:24.020","Text":"hence an even function."},{"Start":"02:24.020 ","End":"02:28.610","Text":"We can just double this and take it from 0 to Pi."},{"Start":"02:28.610 ","End":"02:37.160","Text":"The integral of sine is minus cosine nx over n. Put in the limits and as usual,"},{"Start":"02:37.160 ","End":"02:40.670","Text":"we separate the evens from the odds."},{"Start":"02:40.670 ","End":"02:43.775","Text":"The evens come out to be 0."},{"Start":"02:43.775 ","End":"02:45.650","Text":"For odd n, this is"},{"Start":"02:45.650 ","End":"02:50.420","Text":"the coefficient and so we plug that back in the formula instead of b_n."},{"Start":"02:50.420 ","End":"02:54.350","Text":"We get that g(x) is represented by the following series."},{"Start":"02:54.350 ","End":"02:57.535","Text":"Instead of n, we put 2k minus 1."},{"Start":"02:57.535 ","End":"03:03.079","Text":"Then for f, we just declare that it\u0027s the same expression,"},{"Start":"03:03.079 ","End":"03:06.740","Text":"but only for x between 0 and Pi."},{"Start":"03:06.740 ","End":"03:09.470","Text":"We were asked for the sine series."},{"Start":"03:09.470 ","End":"03:11.945","Text":"Well, this is the sine series of f,"},{"Start":"03:11.945 ","End":"03:17.270","Text":"because f is continuous when will be strictly between 0 and Pi,"},{"Start":"03:17.270 ","End":"03:20.810","Text":"because the function is equal to 1 there certainly continuous."},{"Start":"03:20.810 ","End":"03:24.050","Text":"We can apply the Dirichlet\u0027s theorem and"},{"Start":"03:24.050 ","End":"03:28.130","Text":"say that we have equality here that are the tilt."},{"Start":"03:28.130 ","End":"03:33.140","Text":"We\u0027ve just put an equal sign for x strictly between 0 and Pi."},{"Start":"03:33.140 ","End":"03:36.880","Text":"That completes part A of the exercise."},{"Start":"03:36.880 ","End":"03:40.650","Text":"For part B, we substitute x=Pi over 2."},{"Start":"03:40.650 ","End":"03:44.025","Text":"Note that Pi over 2 is in this interval."},{"Start":"03:44.025 ","End":"03:48.560","Text":"We get just plugging in this constant is the same,"},{"Start":"03:48.560 ","End":"03:52.160","Text":"but here we put x=Pi over 2."},{"Start":"03:52.160 ","End":"03:54.755","Text":"We get that 1 equals this."},{"Start":"03:54.755 ","End":"03:58.725","Text":"Then we computed this before,"},{"Start":"03:58.725 ","End":"04:03.520","Text":"and this comes out to be minus 1^k plus 1."},{"Start":"04:03.520 ","End":"04:05.540","Text":"You can plug in some values and check,"},{"Start":"04:05.540 ","End":"04:07.655","Text":"for example, if k is 1,"},{"Start":"04:07.655 ","End":"04:10.195","Text":"then we get sine of Pi over 2,"},{"Start":"04:10.195 ","End":"04:15.540","Text":"which is minus 1^(1+1), which is 1."},{"Start":"04:15.540 ","End":"04:17.630","Text":"Sine Pi over 2 is 1, and so on."},{"Start":"04:17.630 ","End":"04:19.835","Text":"Anyway, this is what it is."},{"Start":"04:19.835 ","End":"04:24.755","Text":"If we divide by 4 and multiply it by Pi,"},{"Start":"04:24.755 ","End":"04:26.530","Text":"then we get Pi over 4 here,"},{"Start":"04:26.530 ","End":"04:28.805","Text":"the 4 and the Pi disappear from here."},{"Start":"04:28.805 ","End":"04:32.875","Text":"We just get minus 1^(k+1) over 2k minus 1."},{"Start":"04:32.875 ","End":"04:35.730","Text":"Now when k goes 1, 2, 3, 4,"},{"Start":"04:35.730 ","End":"04:37.410","Text":"etc, this goes 1,"},{"Start":"04:37.410 ","End":"04:39.690","Text":"3, 5, 7, etc."},{"Start":"04:39.690 ","End":"04:42.810","Text":"This goes 1 minus 1, 1 minus 1."},{"Start":"04:42.810 ","End":"04:48.020","Text":"It just comes out to be alternating series, this one."},{"Start":"04:48.020 ","End":"04:51.870","Text":"That completes part B and we\u0027re done."}],"ID":28753}],"Thumbnail":null,"ID":294455},{"Name":"Differentiation and Integration of Fourier Series","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Piecewise Continuously Differentable Function","Duration":"4m 10s","ChapterTopicVideoID":27596,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.640","Text":"Now we come to the concept of piecewise continuously differentiable functions."},{"Start":"00:05.640 ","End":"00:08.879","Text":"It\u0027s not directly related to Fourier series,"},{"Start":"00:08.879 ","End":"00:11.520","Text":"but we will need it for the part on"},{"Start":"00:11.520 ","End":"00:16.605","Text":"differentiation of Fourier series which will be in the next clip or so."},{"Start":"00:16.605 ","End":"00:20.700","Text":"Start with an example. Suppose we have the function defined on the interval"},{"Start":"00:20.700 ","End":"00:24.815","Text":"minus 1-1 which is basically the absolute value of x,"},{"Start":"00:24.815 ","End":"00:27.620","Text":"x or minus x accordingly."},{"Start":"00:27.620 ","End":"00:31.085","Text":"Sorry, I got it backwards. Now it\u0027s okay."},{"Start":"00:31.085 ","End":"00:33.890","Text":"Now f is not differentiable at 0,"},{"Start":"00:33.890 ","End":"00:36.410","Text":"it\u0027s got a sharp point there."},{"Start":"00:36.410 ","End":"00:41.535","Text":"However, f prime has both 1-sided limits at 0."},{"Start":"00:41.535 ","End":"00:43.060","Text":"It doesn\u0027t have a 2-sided limit,"},{"Start":"00:43.060 ","End":"00:45.305","Text":"but it has 2 1-sided limits."},{"Start":"00:45.305 ","End":"00:53.510","Text":"The limit as x goes to 0 from the right is the limit of which is 1,"},{"Start":"00:53.510 ","End":"00:57.440","Text":"and the limit from the left is minus 1."},{"Start":"00:57.440 ","End":"01:01.195","Text":"They aren\u0027t equal, but it has both 1-sided limits."},{"Start":"01:01.195 ","End":"01:06.500","Text":"That brings us to the definition of piecewise continuously differentiable."},{"Start":"01:06.500 ","End":"01:12.710","Text":"Function f is said to be piecewise continuously differentiable on a finite interval a,"},{"Start":"01:12.710 ","End":"01:16.805","Text":"b if there is a finite sequence of points going from"},{"Start":"01:16.805 ","End":"01:21.050","Text":"a to b such that the 3 conditions: first of all,"},{"Start":"01:21.050 ","End":"01:26.390","Text":"f is continuously differentiable on each of the sub-intervals x_k, x_k plus 1."},{"Start":"01:26.390 ","End":"01:28.400","Text":"Secondly, at the inner points,"},{"Start":"01:28.400 ","End":"01:30.635","Text":"not counting the first and the last,"},{"Start":"01:30.635 ","End":"01:35.620","Text":"f prime has finite 1-sided limits."},{"Start":"01:35.620 ","End":"01:39.673","Text":"So this is what it is in mathematical language."},{"Start":"01:39.673 ","End":"01:43.730","Text":"Thirdly, at the end points a and b,"},{"Start":"01:43.730 ","End":"01:53.065","Text":"it has to have 1-sided limits respectively from the right at a and from the left at b."},{"Start":"01:53.065 ","End":"01:57.910","Text":"The example we had above of the absolute value of x is piecewise differentiable,"},{"Start":"01:57.910 ","End":"02:00.070","Text":"it satisfies all the conditions."},{"Start":"02:00.070 ","End":"02:02.050","Text":"If we want to formally do it,"},{"Start":"02:02.050 ","End":"02:04.210","Text":"we say that x_naught is minus 1,"},{"Start":"02:04.210 ","End":"02:06.325","Text":"x_1 is 0, x_2 is 1."},{"Start":"02:06.325 ","End":"02:12.896","Text":"It\u0027s continuously differentiable on this open interval from minus 1-0 and also from 0-1."},{"Start":"02:12.896 ","End":"02:14.740","Text":"Minus 1, 0,"},{"Start":"02:14.740 ","End":"02:16.330","Text":"the derivative is minus 1 and on 0,"},{"Start":"02:16.330 ","End":"02:20.980","Text":"1 the derivative is 1,"},{"Start":"02:20.980 ","End":"02:24.760","Text":"and also the 1-sided limits we are okay with that."},{"Start":"02:24.760 ","End":"02:29.578","Text":"From the right at 0 it\u0027s 1,"},{"Start":"02:29.578 ","End":"02:32.915","Text":"and from the left at 0 it\u0027s minus 1."},{"Start":"02:32.915 ","End":"02:34.460","Text":"I forgot to write it,"},{"Start":"02:34.460 ","End":"02:39.540","Text":"but also the 1-sided limit at minus 1 is"},{"Start":"02:39.540 ","End":"02:45.480","Text":"minus 1 and the left limit at 1 is also 1."},{"Start":"02:45.480 ","End":"02:48.270","Text":"Next example. From 0,"},{"Start":"02:48.270 ","End":"02:49.980","Text":"2, the reals,"},{"Start":"02:49.980 ","End":"02:54.650","Text":"f(x) is either x or 3 minus x. Here\u0027s a picture."},{"Start":"02:54.650 ","End":"02:57.275","Text":"Here it\u0027s x, here it\u0027s 3 minus x."},{"Start":"02:57.275 ","End":"02:59.959","Text":"Note that f isn\u0027t continuous,"},{"Start":"02:59.959 ","End":"03:04.400","Text":"nevertheless we\u0027ll see that it is piecewise continuously differentiable."},{"Start":"03:04.400 ","End":"03:07.680","Text":"We divide it up into 0, 1, and 2."},{"Start":"03:07.680 ","End":"03:11.330","Text":"It\u0027s continuously differentiable on 0,"},{"Start":"03:11.330 ","End":"03:14.030","Text":"1 where the derivative is 1,"},{"Start":"03:14.030 ","End":"03:17.330","Text":"and also on the other part the derivative is minus 1,"},{"Start":"03:17.330 ","End":"03:20.990","Text":"and also the 1-sided limit 1 on"},{"Start":"03:20.990 ","End":"03:25.160","Text":"both sides works from the right it\u0027s minus 1 and from the left it\u0027s 1."},{"Start":"03:25.160 ","End":"03:27.065","Text":"Also at the end points,"},{"Start":"03:27.065 ","End":"03:29.570","Text":"the 1-sided limit exists."},{"Start":"03:29.570 ","End":"03:32.185","Text":"At 0 on the right it\u0027s 1,"},{"Start":"03:32.185 ","End":"03:36.525","Text":"and at 2 from the left it\u0027s minus 1."},{"Start":"03:36.525 ","End":"03:39.710","Text":"Now let\u0027s do 1 counter-example of something that"},{"Start":"03:39.710 ","End":"03:42.620","Text":"isn\u0027t piecewise continuously differentiable,"},{"Start":"03:42.620 ","End":"03:46.390","Text":"the square root of x where x goes from 0-2."},{"Start":"03:46.390 ","End":"03:47.930","Text":"It looks fine,"},{"Start":"03:47.930 ","End":"03:49.280","Text":"and smooth, and everything,"},{"Start":"03:49.280 ","End":"03:57.710","Text":"but there\u0027s a problem at 0 because the limit of the derivative as x goes to 0 from"},{"Start":"03:57.710 ","End":"04:02.360","Text":"the right is infinity and we required that the 1-sided limit at"},{"Start":"04:02.360 ","End":"04:07.085","Text":"the endpoints is also finite exists but it\u0027s not infinite,"},{"Start":"04:07.085 ","End":"04:08.662","Text":"so it fails here."},{"Start":"04:08.662 ","End":"04:11.620","Text":"That concludes this clip."}],"ID":28760},{"Watched":false,"Name":"Term by Term Differentiation of Fourier Series","Duration":"6m 34s","ChapterTopicVideoID":27597,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.080","Text":"In this clip, we\u0027ll talk about term by term differentiation of Fourier series."},{"Start":"00:07.080 ","End":"00:10.005","Text":"We\u0027ll start straight away with the theorem."},{"Start":"00:10.005 ","End":"00:13.380","Text":"Let f be a complex valued function on the interval from"},{"Start":"00:13.380 ","End":"00:17.370","Text":"minus pi to pi and it satisfy certain conditions."},{"Start":"00:17.370 ","End":"00:21.765","Text":"First of all, that f is continuous on the interval."},{"Start":"00:21.765 ","End":"00:23.340","Text":"But more than that,"},{"Start":"00:23.340 ","End":"00:28.443","Text":"f at the left end point has to equal f at the right end point."},{"Start":"00:28.443 ","End":"00:35.220","Text":"Which is essentially saying that if we extend f to the whole real line periodically,"},{"Start":"00:35.220 ","End":"00:40.445","Text":"then the periodic extension call it f Tilde is continuous."},{"Start":"00:40.445 ","End":"00:44.660","Text":"If we didn\u0027t have this and it would be a jump discontinuity."},{"Start":"00:44.660 ","End":"00:48.350","Text":"The second requirement on f is that it should be"},{"Start":"00:48.350 ","End":"00:53.015","Text":"piecewise continuously differentiable on this interval."},{"Start":"00:53.015 ","End":"00:56.210","Text":"We\u0027ve just learned what this means."},{"Start":"00:56.210 ","End":"01:03.860","Text":"Then if f is represented by the complex Fourier series this,"},{"Start":"01:03.860 ","End":"01:06.920","Text":"then f prime of x has"},{"Start":"01:06.920 ","End":"01:11.750","Text":"a Fourier series which is gotten by differentiating this term by term."},{"Start":"01:11.750 ","End":"01:15.165","Text":"If we differentiate c_n e^inx,"},{"Start":"01:15.165 ","End":"01:19.250","Text":"then we just get an extra in, in front."},{"Start":"01:19.250 ","End":"01:21.290","Text":"Now let\u0027s prove this,"},{"Start":"01:21.290 ","End":"01:23.720","Text":"not because the proof is that important,"},{"Start":"01:23.720 ","End":"01:28.130","Text":"but because it\u0027s that easy just like an exercise."},{"Start":"01:28.130 ","End":"01:35.750","Text":"First of all note that regarding this line that f and f prime are both in L^2_PC,"},{"Start":"01:35.750 ","End":"01:38.180","Text":"meaning they\u0027re both piecewise continuous,"},{"Start":"01:38.180 ","End":"01:39.965","Text":"f is even continuous."},{"Start":"01:39.965 ","End":"01:41.855","Text":"Of course it\u0027s piecewise continuous,"},{"Start":"01:41.855 ","End":"01:45.470","Text":"and f prime by the definition of piecewise continuously"},{"Start":"01:45.470 ","End":"01:50.090","Text":"differentiable of f means that f prime is piecewise continuous."},{"Start":"01:50.090 ","End":"01:54.960","Text":"They have Fourier series and theorem is just that,"},{"Start":"01:54.960 ","End":"02:00.350","Text":"one can be gotten from the other by term by term differentiation."},{"Start":"02:00.350 ","End":"02:06.140","Text":"The complex nth Fourier coefficient of f prime,"},{"Start":"02:06.140 ","End":"02:08.180","Text":"like the corresponding c_n,"},{"Start":"02:08.180 ","End":"02:14.975","Text":"but for f prime is given by the inner product of f prime with e^inx."},{"Start":"02:14.975 ","End":"02:25.580","Text":"This is equal to 1/2 Pi integral of f prime times the conjugate of e^inx."},{"Start":"02:25.580 ","End":"02:28.160","Text":"We can integrate this by parts,"},{"Start":"02:28.160 ","End":"02:32.780","Text":"you should know if I had the formula already or at least have it handy."},{"Start":"02:32.780 ","End":"02:36.350","Text":"What we get is the following,"},{"Start":"02:36.350 ","End":"02:37.730","Text":"I won\u0027t even read it out."},{"Start":"02:37.730 ","End":"02:40.670","Text":"It\u0027s just standard integration by parts,"},{"Start":"02:40.670 ","End":"02:43.490","Text":"and we evaluate it."},{"Start":"02:43.490 ","End":"02:51.050","Text":"This bit comes out 0 because f of Pi and f of minus Pi are the same,"},{"Start":"02:51.050 ","End":"02:55.700","Text":"and e^minus in Pi is the same as e^in Pi."},{"Start":"02:55.700 ","End":"02:58.550","Text":"Each of them is equal to minus 1^n."},{"Start":"02:58.550 ","End":"03:01.045","Text":"Suppose subtracting the same thing,"},{"Start":"03:01.045 ","End":"03:05.705","Text":"what we\u0027re left with is the integral of this times this."},{"Start":"03:05.705 ","End":"03:13.715","Text":"Then we can pull the in in front together with the minus makes it plus 1/2 Pi was here."},{"Start":"03:13.715 ","End":"03:19.265","Text":"What we\u0027re left with is f of x e^minus inx, dx."},{"Start":"03:19.265 ","End":"03:23.060","Text":"This is just the formula for c_n."},{"Start":"03:23.060 ","End":"03:27.785","Text":"You could write it as the inner product of f with e^inx,"},{"Start":"03:27.785 ","End":"03:29.165","Text":"just like we did here."},{"Start":"03:29.165 ","End":"03:35.780","Text":"The integral of something times e^minus in x is the inner product of this with e^inx."},{"Start":"03:35.780 ","End":"03:39.050","Text":"Anyway, we got the result we wanted from the theorem,"},{"Start":"03:39.050 ","End":"03:47.079","Text":"that the series for this just looks like this with an extra in in front."},{"Start":"03:47.079 ","End":"03:50.385","Text":"That concludes the proof and now a remark."},{"Start":"03:50.385 ","End":"03:52.565","Text":"We did the complex case,"},{"Start":"03:52.565 ","End":"03:58.040","Text":"but term by term differentiation also worked for real Fourier expansion."},{"Start":"03:58.040 ","End":"04:02.600","Text":"If we had that f has this Fourier series,"},{"Start":"04:02.600 ","End":"04:06.380","Text":"then f prime you just get by differentiating the constant disappears,"},{"Start":"04:06.380 ","End":"04:10.610","Text":"cosine nx becomes minus n sine nx."},{"Start":"04:10.610 ","End":"04:17.944","Text":"That\u0027s this minus and the n and the sine nx and the derivative of sine nx is cosine nx."},{"Start":"04:17.944 ","End":"04:20.960","Text":"This bn goes here with an n in front of it."},{"Start":"04:20.960 ","End":"04:23.300","Text":"Yeah, works with reals also."},{"Start":"04:23.300 ","End":"04:25.475","Text":"Now, an example."},{"Start":"04:25.475 ","End":"04:31.235","Text":"We showed in previous clips that the absolute value of x,"},{"Start":"04:31.235 ","End":"04:33.020","Text":"really x in bars,"},{"Start":"04:33.020 ","End":"04:36.715","Text":"is represented by this Fourier series."},{"Start":"04:36.715 ","End":"04:41.960","Text":"We also showed that the signum function signum with a g,"},{"Start":"04:41.960 ","End":"04:47.200","Text":"signum, is represented by this series,"},{"Start":"04:47.200 ","End":"04:49.768","Text":"which is somewhat similar to this."},{"Start":"04:49.768 ","End":"04:52.300","Text":"There\u0027s a connection between these two."},{"Start":"04:52.300 ","End":"04:57.280","Text":"The signum is the derivative of the absolute value function."},{"Start":"04:57.280 ","End":"04:58.659","Text":"See it in a picture."},{"Start":"04:58.659 ","End":"05:03.630","Text":"Over here, the derivative is 1."},{"Start":"05:03.630 ","End":"05:06.945","Text":"It has a constant slope there."},{"Start":"05:06.945 ","End":"05:12.010","Text":"Over here, the slope is minus 1,"},{"Start":"05:12.010 ","End":"05:16.520","Text":"so the derivative is minus 1."},{"Start":"05:16.520 ","End":"05:20.500","Text":"This is the derivative of this at least piecewise."},{"Start":"05:20.500 ","End":"05:24.780","Text":"We\u0027d expect to be able to use term wise differentiation."},{"Start":"05:24.780 ","End":"05:28.085","Text":"Let\u0027s see, if we take typical term here."},{"Start":"05:28.085 ","End":"05:30.440","Text":"Well, the Pi/2 is a constant we don\u0027t need that,"},{"Start":"05:30.440 ","End":"05:31.610","Text":"it goes to 0."},{"Start":"05:31.610 ","End":"05:35.540","Text":"But if we differentiate this part here,"},{"Start":"05:35.540 ","End":"05:41.990","Text":"what we get is the cosine becomes minus sine."},{"Start":"05:41.990 ","End":"05:46.639","Text":"Also we get this constant is coefficient of x coming out in front."},{"Start":"05:46.639 ","End":"05:51.410","Text":"This will cancel out with the squared over here."},{"Start":"05:51.410 ","End":"05:58.115","Text":"What it will leave us is 4/Pi 2k minus 1 sine of this,"},{"Start":"05:58.115 ","End":"06:02.345","Text":"is also the minus cancels with the minus here."},{"Start":"06:02.345 ","End":"06:07.775","Text":"This is exactly what we have over here."},{"Start":"06:07.775 ","End":"06:13.820","Text":"This is a verification of the theorem on term by term differentiation."},{"Start":"06:13.820 ","End":"06:18.050","Text":"I guess I should have said that the absolute value is"},{"Start":"06:18.050 ","End":"06:23.719","Text":"continuous on this interval and the end points are equal."},{"Start":"06:23.719 ","End":"06:25.490","Text":"The value of the function at the endpoints,"},{"Start":"06:25.490 ","End":"06:30.395","Text":"which is what we needed for applying the term by term differentiation."},{"Start":"06:30.395 ","End":"06:34.890","Text":"Everything\u0027s okay and we are done."}],"ID":28761},{"Watched":false,"Name":"Term by Term Integration of Fourier Series","Duration":"6m ","ChapterTopicVideoID":27598,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.260 ","End":"00:05.650","Text":"We\u0027ve just done term by term differentiation of Fourier series."},{"Start":"00:05.650 ","End":"00:09.760","Text":"Now we\u0027ll do term by term integration of Fourier series."},{"Start":"00:09.760 ","End":"00:11.770","Text":"It\u0027s a bit different."},{"Start":"00:11.770 ","End":"00:13.450","Text":"One thing i, n the theorem,"},{"Start":"00:13.450 ","End":"00:17.770","Text":"there\u0027s less requirement on f. All we need is for"},{"Start":"00:17.770 ","End":"00:22.975","Text":"f to be piece-wise continuous on minus pi, pi."},{"Start":"00:22.975 ","End":"00:31.685","Text":"Let\u0027s say it\u0027s represented with Fourier series complex as sum of C_n e to the inx."},{"Start":"00:31.685 ","End":"00:35.380","Text":"Then we can integrate both sides."},{"Start":"00:35.380 ","End":"00:42.405","Text":"Let\u0027s make it a definite integral from minus pi to x of f(t) dt."},{"Start":"00:42.405 ","End":"00:46.735","Text":"Then here we get c_0(x plus pi)."},{"Start":"00:46.735 ","End":"00:51.380","Text":"This comes from the constant term and all the others turn out to be"},{"Start":"00:51.380 ","End":"00:57.320","Text":"C_n over in e^inx minus minus 1^n."},{"Start":"00:57.320 ","End":"01:05.255","Text":"That comes from the integral from minus pi to x and minus pi,"},{"Start":"01:05.255 ","End":"01:08.780","Text":"the e^inx is minus 1^n."},{"Start":"01:08.780 ","End":"01:12.380","Text":"There is a variation on this where we don\u0027t have to put the constants in,"},{"Start":"01:12.380 ","End":"01:15.967","Text":"you don\u0027t have to put the minus 1^n or the c_0 pi here,"},{"Start":"01:15.967 ","End":"01:20.675","Text":"but then we have to add a constant to some side and figure out what it is."},{"Start":"01:20.675 ","End":"01:23.120","Text":"You\u0027ll see that in the example exercise."},{"Start":"01:23.120 ","End":"01:28.145","Text":"Now, the thing is that although it\u0027s term by term integration,"},{"Start":"01:28.145 ","End":"01:30.695","Text":"and although we started with a Fourier series,"},{"Start":"01:30.695 ","End":"01:34.820","Text":"this is not a Fourier series because of the x here."},{"Start":"01:34.820 ","End":"01:38.495","Text":"C_0 x is not part of a Fourier series."},{"Start":"01:38.495 ","End":"01:45.570","Text":"The constant is part of it because the constant is like something e^i0x."},{"Start":"01:46.210 ","End":"01:54.920","Text":"It\u0027s not. However, we do get that it converges uniformly on the interval,"},{"Start":"01:54.920 ","End":"01:56.456","Text":"which implies, in particular,"},{"Start":"01:56.456 ","End":"01:58.880","Text":"that it converges point-wise."},{"Start":"01:58.880 ","End":"02:00.380","Text":"That\u0027s the first remark."},{"Start":"02:00.380 ","End":"02:05.030","Text":"Second remark is just like in the case of differentiation,"},{"Start":"02:05.030 ","End":"02:08.480","Text":"it also works for the real case."},{"Start":"02:08.480 ","End":"02:13.115","Text":"In the real case when we have cosines and sines,"},{"Start":"02:13.115 ","End":"02:17.180","Text":"what we get for the integral is the following."},{"Start":"02:17.180 ","End":"02:22.070","Text":"Again, we have a term containing x which shows that it\u0027s not a Fourier series."},{"Start":"02:22.070 ","End":"02:23.825","Text":"Let\u0027s do an example."},{"Start":"02:23.825 ","End":"02:30.035","Text":"Before that, I forgot to say that this equals here is not a mistake."},{"Start":"02:30.035 ","End":"02:37.110","Text":"The equals comes from the fact that we have uniform convergence."},{"Start":"02:37.110 ","End":"02:38.675","Text":"In the example,"},{"Start":"02:38.675 ","End":"02:43.275","Text":"we\u0027ll be working on the interval 0 to 2 pi instead of the usual minus pi pi."},{"Start":"02:43.275 ","End":"02:44.840","Text":"It\u0027s almost the same thing."},{"Start":"02:44.840 ","End":"02:52.354","Text":"We\u0027re given the function pi minus x/2 has a Fourier series as follows."},{"Start":"02:52.354 ","End":"02:55.025","Text":"You could do it as an exercise on your own."},{"Start":"02:55.025 ","End":"02:58.220","Text":"You should be able to do such an exercise at this stage."},{"Start":"02:58.220 ","End":"03:00.980","Text":"Anyway, we have to take this as given,"},{"Start":"03:00.980 ","End":"03:05.345","Text":"and use term by term integration to find g,"},{"Start":"03:05.345 ","End":"03:08.540","Text":"where g(x) is a similar series,"},{"Start":"03:08.540 ","End":"03:14.520","Text":"but with an n cubed in the denominator instead of the n. Start with"},{"Start":"03:14.520 ","End":"03:21.410","Text":"this and then do a term by term integration."},{"Start":"03:21.410 ","End":"03:25.955","Text":"The sine becomes minus cosine."},{"Start":"03:25.955 ","End":"03:32.750","Text":"We have to divide by n and we need to add a constant of integration,"},{"Start":"03:32.750 ","End":"03:35.840","Text":"which we\u0027ll do as a_0 over 2."},{"Start":"03:35.840 ","End":"03:40.700","Text":"Reason is what we get here is a Fourier series this time."},{"Start":"03:40.700 ","End":"03:42.410","Text":"Because there was no constant term here,"},{"Start":"03:42.410 ","End":"03:44.470","Text":"so we don\u0027t get that x term."},{"Start":"03:44.470 ","End":"03:46.940","Text":"Then we have a formula for Fourier series,"},{"Start":"03:46.940 ","End":"03:48.110","Text":"what the a_0 is."},{"Start":"03:48.110 ","End":"03:50.135","Text":"It\u0027s convenient to do it this way."},{"Start":"03:50.135 ","End":"03:52.880","Text":"I should\u0027ve said I\u0027m doing an indefinite integral this time."},{"Start":"03:52.880 ","End":"03:55.744","Text":"That\u0027s why we had to add a constant to one of the sides."},{"Start":"03:55.744 ","End":"04:02.255","Text":"The integral of the left-hand side is pi over 2x minus x^2/ 4."},{"Start":"04:02.255 ","End":"04:06.070","Text":"The right-hand side just leaving as it is."},{"Start":"04:06.070 ","End":"04:09.640","Text":"Put equals here because we know that"},{"Start":"04:09.640 ","End":"04:15.100","Text":"the integration Fourier series is uniformly convergent."},{"Start":"04:15.100 ","End":"04:17.500","Text":"Even if it isn\u0027t Fourier, it\u0027s uniform convergence."},{"Start":"04:17.500 ","End":"04:21.145","Text":"We can write equalities instead of just the tilled."},{"Start":"04:21.145 ","End":"04:29.110","Text":"Then from here, we can compute a_0 using the formula for a_0 that we know,"},{"Start":"04:29.110 ","End":"04:33.120","Text":"except that we put 0 to 2 pi instead of minus pi,"},{"Start":"04:33.120 ","End":"04:35.380","Text":"pi, which is what we\u0027re more used to,"},{"Start":"04:35.380 ","End":"04:37.884","Text":"but it makes no difference, any 2 pi interval."},{"Start":"04:37.884 ","End":"04:40.660","Text":"This is a straightforward integration."},{"Start":"04:40.660 ","End":"04:41.830","Text":"What we get is this,"},{"Start":"04:41.830 ","End":"04:43.707","Text":"but from 0 to 2 pi."},{"Start":"04:43.707 ","End":"04:47.210","Text":"When we plug in 0, we get nothing."},{"Start":"04:47.210 ","End":"04:49.580","Text":"When we plug in 2 pi here we got 4 pi squared,"},{"Start":"04:49.580 ","End":"04:51.800","Text":"here 8 pi cubed."},{"Start":"04:51.800 ","End":"04:56.560","Text":"This boils down to 1/3 pi squared. That\u0027s a_0."},{"Start":"04:56.560 ","End":"04:58.780","Text":"We can put it back in here,"},{"Start":"04:58.780 ","End":"05:02.120","Text":"a_0/2 is pi squared over 6."},{"Start":"05:02.120 ","End":"05:03.775","Text":"This is what we get."},{"Start":"05:03.775 ","End":"05:06.340","Text":"Now, throw the series to the left and everything"},{"Start":"05:06.340 ","End":"05:10.550","Text":"else on the right and this is what we have."},{"Start":"05:11.240 ","End":"05:15.715","Text":"The plan now is to do another term wise integration."},{"Start":"05:15.715 ","End":"05:21.130","Text":"This time we\u0027ll get sine nx and we\u0027ll also have n cubed in the denominator."},{"Start":"05:21.130 ","End":"05:24.309","Text":"Doing that, we get the following,"},{"Start":"05:24.309 ","End":"05:26.670","Text":"sine nx over n^3."},{"Start":"05:26.670 ","End":"05:32.090","Text":"Here, just the straightforward integral."},{"Start":"05:32.090 ","End":"05:34.250","Text":"We don\u0027t need the constant here."},{"Start":"05:34.250 ","End":"05:35.964","Text":"Well, there is a constant. It\u0027s 0."},{"Start":"05:35.964 ","End":"05:37.700","Text":"Because when we plug in 0,"},{"Start":"05:37.700 ","End":"05:40.306","Text":"all the sines are 0 and here x, x^2,"},{"Start":"05:40.306 ","End":"05:42.020","Text":"and x^3 are all zeros,"},{"Start":"05:42.020 ","End":"05:45.690","Text":"so we don\u0027t need any additional constant."},{"Start":"05:45.690 ","End":"05:48.965","Text":"The answer is that g(x),"},{"Start":"05:48.965 ","End":"05:51.155","Text":"which was this series,"},{"Start":"05:51.155 ","End":"05:53.270","Text":"is just copying from here,"},{"Start":"05:53.270 ","End":"06:00.570","Text":"pi squared over 6(x) minus pi over 4 (x^2) plus x^3/12 and we are done."}],"ID":28762},{"Watched":false,"Name":"Exercise 1","Duration":"4m 41s","ChapterTopicVideoID":27599,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.945","Text":"In this exercise, we\u0027re going to prove the following proposition."},{"Start":"00:03.945 ","End":"00:11.580","Text":"Let f be piecewise continuously differentiable on interval minus Pi to Pi,"},{"Start":"00:11.580 ","End":"00:15.390","Text":"such that f of Pi is f of minus Pi,"},{"Start":"00:15.390 ","End":"00:17.565","Text":"meaning it\u0027s equal at the endpoints,"},{"Start":"00:17.565 ","End":"00:22.155","Text":"then the complex Fourier series converges absolutely."},{"Start":"00:22.155 ","End":"00:28.415","Text":"We call that piecewise continuously differentiable means that it\u0027s differentiable,"},{"Start":"00:28.415 ","End":"00:31.640","Text":"but before that it\u0027s continuous of course then it\u0027s differentiable"},{"Start":"00:31.640 ","End":"00:35.435","Text":"and the derivative is piecewise continuous."},{"Start":"00:35.435 ","End":"00:42.485","Text":"Let\u0027s say that the complex Fourier series for f is the sum c n e^inx,"},{"Start":"00:42.485 ","End":"00:44.660","Text":"from minus infinity to infinity."},{"Start":"00:44.660 ","End":"00:49.580","Text":"What we have to do is to prove that this series converges absolutely,"},{"Start":"00:49.580 ","End":"00:55.520","Text":"and what this means is that the series of the absolute value converges."},{"Start":"00:55.520 ","End":"00:57.025","Text":"First prove that."},{"Start":"00:57.025 ","End":"01:04.475","Text":"What\u0027s going to help us here is the theorem on term by term differentiation."},{"Start":"01:04.475 ","End":"01:08.915","Text":"In the theorem we have a function which is continuous"},{"Start":"01:08.915 ","End":"01:13.190","Text":"and f of minus Pi the f of Pi meaning equal at the endpoints."},{"Start":"01:13.190 ","End":"01:17.090","Text":"Well, that\u0027s what we have for our function f. Yeah,"},{"Start":"01:17.090 ","End":"01:24.770","Text":"there it is and we have that it\u0027s piecewise continuously differentiable, so yes."},{"Start":"01:24.770 ","End":"01:27.799","Text":"we\u0027ve satisfied the conditions of this theorem."},{"Start":"01:27.799 ","End":"01:33.620","Text":"We can conclude that if this is the complex Fourier series for f,"},{"Start":"01:33.620 ","End":"01:36.440","Text":"then we can differentiate term by term,"},{"Start":"01:36.440 ","End":"01:41.750","Text":"meaning that f prime is incne^inx,"},{"Start":"01:41.750 ","End":"01:44.590","Text":"sum from minus infinity to infinity."},{"Start":"01:44.590 ","End":"01:47.445","Text":"Okay so just write that out."},{"Start":"01:47.445 ","End":"01:53.644","Text":"In our case this holds and let\u0027s apply Parseval\u0027s theorem to f prime."},{"Start":"01:53.644 ","End":"01:59.990","Text":"What we get is that the sum of each term absolute value squared,"},{"Start":"01:59.990 ","End":"02:03.500","Text":"is 1 over 2 pi the integral"},{"Start":"02:03.500 ","End":"02:08.320","Text":"of the absolute value of the function squared only the function here is f prime."},{"Start":"02:08.320 ","End":"02:10.580","Text":"Doesn\u0027t matter what it converges to."},{"Start":"02:10.580 ","End":"02:14.540","Text":"The point is that it converges and we can rewrite this because i squared,"},{"Start":"02:14.540 ","End":"02:20.275","Text":"the absolute value is 1 and absolute value of n squared is just n squared."},{"Start":"02:20.275 ","End":"02:24.900","Text":"We get that this is equal to this, so this converges."},{"Start":"02:26.180 ","End":"02:34.455","Text":"From this, what we can get is that the sum of the cn\u0027s,"},{"Start":"02:34.455 ","End":"02:38.210","Text":"but excluding the 0 term is equal"},{"Start":"02:38.210 ","End":"02:43.170","Text":"to the sum of absolute values that of c n we can write 1 over n(n),"},{"Start":"02:43.170 ","End":"02:47.165","Text":"and that\u0027s why we need the n not equal to 0."},{"Start":"02:47.165 ","End":"02:50.780","Text":"This Is less than or equal to k. I need to explain,"},{"Start":"02:50.780 ","End":"02:53.075","Text":"going to use any of the inequality here,"},{"Start":"02:53.075 ","End":"02:56.720","Text":"that if x and y are real numbers,"},{"Start":"02:56.720 ","End":"02:57.980","Text":"then absolute value of x,"},{"Start":"02:57.980 ","End":"03:02.570","Text":"y is less than or equal to a half of (x^2+y^2)."},{"Start":"03:02.570 ","End":"03:05.840","Text":"That\u0027s easy to prove if you just take this expression"},{"Start":"03:05.840 ","End":"03:09.765","Text":"and expand it a bit and collect terms and so on."},{"Start":"03:09.765 ","End":"03:11.645","Text":"It\u0027s an easy exercise,"},{"Start":"03:11.645 ","End":"03:15.110","Text":"or you can just say this is a well-known inequality."},{"Start":"03:15.110 ","End":"03:18.335","Text":"I\u0027ve color-coded it to help us see what\u0027s what."},{"Start":"03:18.335 ","End":"03:20.825","Text":"This is the x, this is the y."},{"Start":"03:20.825 ","End":"03:26.860","Text":"We get this as less than or equal to a half of (x^2+y^2) is this."},{"Start":"03:26.860 ","End":"03:30.490","Text":"Break this up into 2 sums,"},{"Start":"03:30.490 ","End":"03:36.350","Text":"which is justified because each one of them converges is finite."},{"Start":"03:36.350 ","End":"03:39.335","Text":"1 over n squared is a famous series."},{"Start":"03:39.335 ","End":"03:41.810","Text":"It actually converges to Pi squared over 6,"},{"Start":"03:41.810 ","End":"03:43.420","Text":"but never mind,"},{"Start":"03:43.420 ","End":"03:47.940","Text":"and this just shown here it converges to something."},{"Start":"03:47.940 ","End":"03:51.344","Text":"Convergent plus convergent is convergent,"},{"Start":"03:51.344 ","End":"03:55.460","Text":"and the only thing we need to fix up is this not equal to 0,"},{"Start":"03:55.460 ","End":"03:57.305","Text":"but that\u0027s just one term."},{"Start":"03:57.305 ","End":"04:01.057","Text":"We can add that term just excluded the 0 term,"},{"Start":"04:01.057 ","End":"04:04.765","Text":"so add it back on it doesn\u0027t change convergence if you add one term."},{"Start":"04:04.765 ","End":"04:07.070","Text":"We\u0027ve proved that this sum of"},{"Start":"04:07.070 ","End":"04:10.775","Text":"absolute value of cn is less than infinity, meaning converges."},{"Start":"04:10.775 ","End":"04:15.810","Text":"The sum that we wanted is this one, cne^inx,"},{"Start":"04:15.810 ","End":"04:17.480","Text":"or we can split this up as"},{"Start":"04:17.480 ","End":"04:22.310","Text":"a product absolute value of cn absolute value of e^inx and this is one."},{"Start":"04:22.310 ","End":"04:26.015","Text":"This is just the sum of the absolute value of cn"},{"Start":"04:26.015 ","End":"04:30.095","Text":"which from here converges and so this converges,"},{"Start":"04:30.095 ","End":"04:34.700","Text":"which is what we wanted to show here like we said, if this converges,"},{"Start":"04:34.700 ","End":"04:37.880","Text":"it means that this converges absolutely,"},{"Start":"04:37.880 ","End":"04:40.860","Text":"and that concludes this exercise."}],"ID":28763},{"Watched":false,"Name":"Exercise 2 - Part a","Duration":"4m 29s","ChapterTopicVideoID":27600,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.675","Text":"This exercise has 6 parts,"},{"Start":"00:03.675 ","End":"00:06.810","Text":"mostly concerning this function,"},{"Start":"00:06.810 ","End":"00:10.275","Text":"which is an inverse parabola,"},{"Start":"00:10.275 ","End":"00:16.305","Text":"x times Pi minus x on the interval from 0 to Pi."},{"Start":"00:16.305 ","End":"00:18.528","Text":"Read each part as we come to it."},{"Start":"00:18.528 ","End":"00:22.995","Text":"Part a, you want to expand f as a sine series."},{"Start":"00:22.995 ","End":"00:25.500","Text":"If it\u0027s going to be a sine series,"},{"Start":"00:25.500 ","End":"00:27.900","Text":"it\u0027s going to be an odd function."},{"Start":"00:27.900 ","End":"00:29.430","Text":"What we\u0027ll do is we\u0027ll,"},{"Start":"00:29.430 ","End":"00:31.710","Text":"first of all,"},{"Start":"00:31.710 ","End":"00:38.265","Text":"extend the function little f to a function big F and all of minus by Pi."},{"Start":"00:38.265 ","End":"00:43.415","Text":"Usual formula, we take it as the same as f on"},{"Start":"00:43.415 ","End":"00:49.430","Text":"0 to Pi and minus f of minus x from minus Pi to 0."},{"Start":"00:49.430 ","End":"00:52.180","Text":"F is an odd function."},{"Start":"00:52.180 ","End":"00:55.560","Text":"It only has sines and no cosines."},{"Start":"00:55.560 ","End":"00:59.010","Text":"It\u0027s of the form the sum of bn sine nx."},{"Start":"00:59.010 ","End":"01:05.210","Text":"We have to compute the bn and there\u0027s a formula for it, the usual formula."},{"Start":"01:05.210 ","End":"01:10.940","Text":"Because f is equal to little f on 0 to Pi and this is odd,"},{"Start":"01:10.940 ","End":"01:15.920","Text":"we can just say that this is twice the integral just from 0 to Pi."},{"Start":"01:15.920 ","End":"01:17.300","Text":"Then when we found the bn,"},{"Start":"01:17.300 ","End":"01:20.930","Text":"we can say that little f has the same series as big F."},{"Start":"01:20.930 ","End":"01:25.475","Text":"We just have to restrict x to the interval 0 Pi."},{"Start":"01:25.475 ","End":"01:27.410","Text":"Now let\u0027s compute bn."},{"Start":"01:27.410 ","End":"01:31.730","Text":"This is the formula and it\u0027s going to be quite technical."},{"Start":"01:31.730 ","End":"01:34.400","Text":"We\u0027ll use integration by parts."},{"Start":"01:34.400 ","End":"01:35.720","Text":"Break it up, first of all,"},{"Start":"01:35.720 ","End":"01:37.655","Text":"expand this. Now, break it up."},{"Start":"01:37.655 ","End":"01:43.145","Text":"This will be the part we differentiate and this is the part we integrate."},{"Start":"01:43.145 ","End":"01:47.780","Text":"What we get is this times g,"},{"Start":"01:47.780 ","End":"01:49.490","Text":"which is the integral of this,"},{"Start":"01:49.490 ","End":"01:55.450","Text":"which is minus cosine nx over n from 0 to Pi,"},{"Start":"01:55.450 ","End":"02:02.120","Text":"minus the integral of this f times the g we found,"},{"Start":"02:02.120 ","End":"02:05.360","Text":"which is this from 0 to Pi."},{"Start":"02:05.360 ","End":"02:08.555","Text":"The minus and the minus will cancel"},{"Start":"02:08.555 ","End":"02:13.190","Text":"and we\u0027ll take the n in front of the integral area and all of"},{"Start":"02:13.190 ","End":"02:21.980","Text":"this part comes out to 0 because this is 0 when x is Pi or 0."},{"Start":"02:21.980 ","End":"02:30.775","Text":"What we end up with is 2 over n Pi times this part here without the minus."},{"Start":"02:30.775 ","End":"02:34.050","Text":"Again, we\u0027ll use integration by parts,"},{"Start":"02:34.050 ","End":"02:36.005","Text":"same formula of this time."},{"Start":"02:36.005 ","End":"02:37.970","Text":"This will be what we differentiate,"},{"Start":"02:37.970 ","End":"02:40.085","Text":"this is what we integrate."},{"Start":"02:40.085 ","End":"02:47.330","Text":"Very similarly, we get this expression and again the minuses cancel."},{"Start":"02:47.330 ","End":"02:50.990","Text":"We bring the 2 in front as well as the n,"},{"Start":"02:50.990 ","End":"02:56.390","Text":"we get 4 over n squared Pi times the integral of sine nx,"},{"Start":"02:56.390 ","End":"02:59.015","Text":"dx from 0 to Pi,"},{"Start":"02:59.015 ","End":"03:07.070","Text":"integral of sine is minus cosine nx over n. Then we need to evaluate it between 0 and Pi."},{"Start":"03:07.070 ","End":"03:12.070","Text":"We can get rid of the minus and change the order of these two."},{"Start":"03:12.070 ","End":"03:18.200","Text":"Then when x is 0, we get cosine of 0 is 1 and when x is Pi,"},{"Start":"03:18.200 ","End":"03:19.670","Text":"we have cosine of n Pi,"},{"Start":"03:19.670 ","End":"03:24.270","Text":"which we know is minus 1 to the n. This is what we get."},{"Start":"03:24.270 ","End":"03:27.089","Text":"Then when n is odd,"},{"Start":"03:27.089 ","End":"03:29.460","Text":"so 2k minus 1,"},{"Start":"03:29.460 ","End":"03:33.030","Text":"then this will be 1 minus minus 1 is 2,"},{"Start":"03:33.030 ","End":"03:35.000","Text":"so 8 over n^3 Pi,"},{"Start":"03:35.000 ","End":"03:41.525","Text":"and when n is even we\u0027ll get 1 minus 1 here is 0 so this is our expression for bn."},{"Start":"03:41.525 ","End":"03:48.815","Text":"Now, we just have to throw it into the formula for f. The sum of bn sine nx."},{"Start":"03:48.815 ","End":"03:51.440","Text":"Like I said, little f will be the same thing"},{"Start":"03:51.440 ","End":"03:54.940","Text":"except that we will consider it only from 0 to Pi."},{"Start":"03:54.940 ","End":"03:57.060","Text":"Then put the bn,"},{"Start":"03:57.060 ","End":"04:02.610","Text":"but bn is only non 0 when n is equal to 2k minus 1."},{"Start":"04:02.610 ","End":"04:05.820","Text":"We get the sum of k from 1 to infinity,"},{"Start":"04:05.820 ","End":"04:13.800","Text":"replace the n here by 2k minus 1,"},{"Start":"04:13.800 ","End":"04:19.230","Text":"and the n here also 2k minus 1."},{"Start":"04:19.230 ","End":"04:26.180","Text":"This is our formula for the Fourier expansion of x times Pi minus x."},{"Start":"04:26.180 ","End":"04:29.340","Text":"That concludes part a."}],"ID":28764},{"Watched":false,"Name":"Exercise 2 - Part b","Duration":"2m 15s","ChapterTopicVideoID":27601,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"Continuing with this exercise,"},{"Start":"00:02.280 ","End":"00:04.290","Text":"we just finished a, now part b;"},{"Start":"00:04.290 ","End":"00:07.515","Text":"to what function does the series converge"},{"Start":"00:07.515 ","End":"00:12.495","Text":"and sketch the graph of this function for at least 3 period."},{"Start":"00:12.495 ","End":"00:13.970","Text":"I remember in part a,"},{"Start":"00:13.970 ","End":"00:20.795","Text":"we extended the graph of x times pi minus x to an odd function."},{"Start":"00:20.795 ","End":"00:24.365","Text":"It looks like a sine function, but it isn\u0027t."},{"Start":"00:24.365 ","End":"00:27.815","Text":"We took f(x) which is x pi minus x,"},{"Start":"00:27.815 ","End":"00:33.845","Text":"and rotated it 180 degrees to make it odd minus f of minus x."},{"Start":"00:33.845 ","End":"00:37.430","Text":"If you plug in instead of x minus x,"},{"Start":"00:37.430 ","End":"00:39.035","Text":"you\u0027ll get here plus x."},{"Start":"00:39.035 ","End":"00:42.545","Text":"Here, we\u0027ll get minus x when we take the minus,"},{"Start":"00:42.545 ","End":"00:45.230","Text":"becomes x pi plus x."},{"Start":"00:45.230 ","End":"00:48.170","Text":"This is what it is from 0 to pi,"},{"Start":"00:48.170 ","End":"00:52.710","Text":"this is what it is from minus pi to 0 because this function"},{"Start":"00:52.710 ","End":"00:57.605","Text":"is continuous by the [inaudible] theorem,"},{"Start":"00:57.605 ","End":"01:04.150","Text":"the series actually converges to the function at least from minus pi to pi."},{"Start":"01:04.150 ","End":"01:08.920","Text":"But because this is periodic and has period 2pi,"},{"Start":"01:08.920 ","End":"01:11.485","Text":"that\u0027s how we extend it."},{"Start":"01:11.485 ","End":"01:15.490","Text":"All the sign and x period 2pi or less,"},{"Start":"01:15.490 ","End":"01:22.270","Text":"but they also have a period 2pi so we extend it with 2pi period and you know what,"},{"Start":"01:22.270 ","End":"01:24.355","Text":"I\u0027ll show the picture first."},{"Start":"01:24.355 ","End":"01:28.165","Text":"Here\u0027s what it is. If we extend this function,"},{"Start":"01:28.165 ","End":"01:32.050","Text":"I took another period on the left and other period on the right."},{"Start":"01:32.050 ","End":"01:35.425","Text":"This is the original x pi minus x,"},{"Start":"01:35.425 ","End":"01:39.125","Text":"this is what it looks like and it is not a sine wave."},{"Start":"01:39.125 ","End":"01:42.290","Text":"It is this replicated here,"},{"Start":"01:42.290 ","End":"01:48.215","Text":"and then this part is copy paste infinitely."},{"Start":"01:48.215 ","End":"01:50.795","Text":"This would be the formula."},{"Start":"01:50.795 ","End":"01:55.400","Text":"If we figure it, we just replace x by x minus"},{"Start":"01:55.400 ","End":"02:01.375","Text":"2k pi and then the range will also shift by 2k pi."},{"Start":"02:01.375 ","End":"02:04.280","Text":"This would be the formula and give it a different name,"},{"Start":"02:04.280 ","End":"02:06.830","Text":"f with a hat over it OF carrot"},{"Start":"02:06.830 ","End":"02:10.625","Text":"to say this is the function that\u0027s been extended to all of the reals."},{"Start":"02:10.625 ","End":"02:15.240","Text":"I think that\u0027s enough said on that concludes Part B."}],"ID":28765},{"Watched":false,"Name":"Exercise 2 - Part c","Duration":"3m 8s","ChapterTopicVideoID":27602,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:03.180","Text":"Now, part c of this exercise,"},{"Start":"00:03.180 ","End":"00:06.030","Text":"and we\u0027ll use the result of part a."},{"Start":"00:06.030 ","End":"00:10.830","Text":"We have to prove the sum of this series. Won\u0027t read it out."},{"Start":"00:10.830 ","End":"00:13.440","Text":"Want to remind you what we have so far."},{"Start":"00:13.440 ","End":"00:17.070","Text":"We extended our function little f, which was this,"},{"Start":"00:17.070 ","End":"00:21.210","Text":"to an odd function on minus Pi to Pi,"},{"Start":"00:21.210 ","End":"00:24.960","Text":"using the formula to make it an odd function."},{"Start":"00:24.960 ","End":"00:31.895","Text":"We showed that the Fourier series for this function big F is the following."},{"Start":"00:31.895 ","End":"00:36.980","Text":"Now, we have to prove this sum of series and there\u0027s 2 main tools,"},{"Start":"00:36.980 ","End":"00:41.075","Text":"either Parseval\u0027s theorem or it\u0027s Dirichlet\u0027s theorem."},{"Start":"00:41.075 ","End":"00:47.450","Text":"The clue that it\u0027s going to be Parseval is because we have the power of 6 here."},{"Start":"00:47.450 ","End":"00:50.795","Text":"If we take the absolute value of a term squared,"},{"Start":"00:50.795 ","End":"00:53.885","Text":"then we get the 1 over 2k minus 1^6."},{"Start":"00:53.885 ","End":"00:55.940","Text":"If we were using the Dirichlet,"},{"Start":"00:55.940 ","End":"00:57.845","Text":"would just have it cubed."},{"Start":"00:57.845 ","End":"01:01.960","Text":"Just repeating what big F is, it\u0027s this."},{"Start":"01:01.960 ","End":"01:07.135","Text":"Here\u0027s a reminder of Parseval\u0027s identity, the real case."},{"Start":"01:07.135 ","End":"01:10.550","Text":"If we have a function on minus Pi to Pi,"},{"Start":"01:10.550 ","End":"01:13.640","Text":"which has the following Fourier series,"},{"Start":"01:13.640 ","End":"01:18.710","Text":"then this is the identity that 1 over pi times the integral of absolute value of f^2,"},{"Start":"01:18.710 ","End":"01:23.570","Text":"is the sum of the squares of the coefficients except for a naught squared,"},{"Start":"01:23.570 ","End":"01:25.495","Text":"which is divided by 2."},{"Start":"01:25.495 ","End":"01:27.970","Text":"In our case we don\u0027t have the a_ns,"},{"Start":"01:27.970 ","End":"01:30.335","Text":"we just have the b_ns because it\u0027s an odd function."},{"Start":"01:30.335 ","End":"01:34.355","Text":"Anyway, this is what we get in our case for Parseval\u0027s theorem."},{"Start":"01:34.355 ","End":"01:36.740","Text":"Now, absolute value of f^2,"},{"Start":"01:36.740 ","End":"01:38.330","Text":"we don\u0027t need the absolute value,"},{"Start":"01:38.330 ","End":"01:42.725","Text":"it\u0027s just f^2, but it\u0027s an even function even though f itself is an odd function."},{"Start":"01:42.725 ","End":"01:46.025","Text":"When it\u0027s squared, it becomes odd times odd is even."},{"Start":"01:46.025 ","End":"01:51.920","Text":"We can take the integral just from 0 to Pi and double it, put a 2 here."},{"Start":"01:51.920 ","End":"01:54.020","Text":"Now here, we can drop the absolute value,"},{"Start":"01:54.020 ","End":"01:59.870","Text":"8^2 is 64, and here we have 2k minus 1^6 Pi^2."},{"Start":"01:59.870 ","End":"02:04.205","Text":"Let\u0027s take the 64 over Pi^2 outside the sum."},{"Start":"02:04.205 ","End":"02:06.664","Text":"This is the sum we need."},{"Start":"02:06.664 ","End":"02:13.715","Text":"Now, let\u0027s flip sides and bring the 64 over Pi^2 to the other side so it\u0027s inverted."},{"Start":"02:13.715 ","End":"02:14.990","Text":"Now this is equal to,"},{"Start":"02:14.990 ","End":"02:16.790","Text":"the Pi^2 over Pi is Pi,"},{"Start":"02:16.790 ","End":"02:19.900","Text":"2/64 is 1/32,"},{"Start":"02:19.900 ","End":"02:23.240","Text":"this if we expand it is the following."},{"Start":"02:23.240 ","End":"02:26.105","Text":"It\u0027s just a straightforward polynomial integral."},{"Start":"02:26.105 ","End":"02:27.825","Text":"This is what it is."},{"Start":"02:27.825 ","End":"02:30.380","Text":"Then substitute 0 we get nothing,"},{"Start":"02:30.380 ","End":"02:32.270","Text":"and we substitute x equals Pi."},{"Start":"02:32.270 ","End":"02:38.005","Text":"All these are Pi^5, Pi^2 Pi^3, Pi^4, Pi^5."},{"Start":"02:38.005 ","End":"02:44.119","Text":"We can take out Pi^5 and we\u0027re left with 1/3 minus 2/4,"},{"Start":"02:44.119 ","End":"02:46.505","Text":"which is 1/2 and plus 1/5."},{"Start":"02:46.505 ","End":"02:48.860","Text":"Take it over a common denominator 30."},{"Start":"02:48.860 ","End":"02:52.895","Text":"I also combine Pi^5 with Pi is Pi^6."},{"Start":"02:52.895 ","End":"02:57.660","Text":"This comes out to be 10 plus 6 minus 15 is 1/30,"},{"Start":"02:57.660 ","End":"03:02.825","Text":"1/30 times 1/32 is 1/960 with the Pi^6."},{"Start":"03:02.825 ","End":"03:09.060","Text":"This gives us what we want and we\u0027ve proved it and that concludes part c."}],"ID":28766},{"Watched":false,"Name":"Exercise 2 - Part d","Duration":"2m 53s","ChapterTopicVideoID":27603,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.520","Text":"Next, we have Part D. Again,"},{"Start":"00:02.520 ","End":"00:04.965","Text":"we have to prove the sum of the series."},{"Start":"00:04.965 ","End":"00:06.210","Text":"In the previous one,"},{"Start":"00:06.210 ","End":"00:09.660","Text":"we had this and we used Parseval."},{"Start":"00:09.660 ","End":"00:12.420","Text":"In Part D we\u0027ll use Dirichlet."},{"Start":"00:12.420 ","End":"00:13.890","Text":"This is what we have to prove."},{"Start":"00:13.890 ","End":"00:15.240","Text":"It has the power of 3,"},{"Start":"00:15.240 ","End":"00:17.655","Text":"but an alternating sign."},{"Start":"00:17.655 ","End":"00:20.280","Text":"Now, we showed,"},{"Start":"00:20.280 ","End":"00:22.020","Text":"just repeating what we have,"},{"Start":"00:22.020 ","End":"00:26.775","Text":"that this is represented by this Fourier series."},{"Start":"00:26.775 ","End":"00:29.790","Text":"We don\u0027t need it on minus Pi."},{"Start":"00:29.790 ","End":"00:33.450","Text":"For this part, we just need it on 0 Pi."},{"Start":"00:33.450 ","End":"00:35.650","Text":"This is the graph."},{"Start":"00:35.650 ","End":"00:38.255","Text":"This is, again, what we want to prove."},{"Start":"00:38.255 ","End":"00:40.955","Text":"How do we get this alternating sign?"},{"Start":"00:40.955 ","End":"00:46.375","Text":"Turns out that the trick is to substitute x equals Pi/2."},{"Start":"00:46.375 ","End":"00:51.785","Text":"It\u0027s almost always one of the endpoints or the midpoint in the Dirichlet theorem."},{"Start":"00:51.785 ","End":"00:56.275","Text":"We can use the theorem because the function is continuous at Pi/2."},{"Start":"00:56.275 ","End":"00:59.620","Text":"What do we get if we substitute Pi/2?"},{"Start":"00:59.620 ","End":"01:03.020","Text":"Well, notice that if we let x equal Pi/2,"},{"Start":"01:03.020 ","End":"01:06.440","Text":"we have sin of 2k minus 1 Pi/2."},{"Start":"01:06.440 ","End":"01:07.715","Text":"Just expand that,"},{"Start":"01:07.715 ","End":"01:10.765","Text":"k Pi minus Pi/2."},{"Start":"01:10.765 ","End":"01:14.460","Text":"We know this to be minus 1 to the k plus 1."},{"Start":"01:14.460 ","End":"01:15.920","Text":"There\u0027s many ways of proving this,"},{"Start":"01:15.920 ","End":"01:19.055","Text":"I\u0027ll just demonstrate it and it works."},{"Start":"01:19.055 ","End":"01:22.060","Text":"That when k=1,"},{"Start":"01:22.060 ","End":"01:27.300","Text":"then we have minus 1^2, which is 1."},{"Start":"01:27.300 ","End":"01:30.300","Text":"It\u0027s true that sine Pi/2 is 1."},{"Start":"01:30.300 ","End":"01:33.000","Text":"That\u0027s what we get when k is 1, Pi minus Pi/2."},{"Start":"01:33.000 ","End":"01:34.980","Text":"But at k=2,"},{"Start":"01:34.980 ","End":"01:36.750","Text":"we have 2 Pi minus Pi/2,"},{"Start":"01:36.750 ","End":"01:38.655","Text":"which is 3 Pi/2."},{"Start":"01:38.655 ","End":"01:43.440","Text":"Here we have minus 1^2 plus 1 is minus 1."},{"Start":"01:43.440 ","End":"01:44.760","Text":"I just check two of them,"},{"Start":"01:44.760 ","End":"01:46.125","Text":"you can check the next one,"},{"Start":"01:46.125 ","End":"01:48.390","Text":"5 Pi/2 is equal to 1,"},{"Start":"01:48.390 ","End":"01:51.360","Text":"and the next one says it\u0027s sine 7 Pi/ 2 is minus 1."},{"Start":"01:51.360 ","End":"01:53.895","Text":"Get the 1 minus 1, 1 minus 1,"},{"Start":"01:53.895 ","End":"01:57.940","Text":"and it\u0027s the minus 1^k plus 1 and we\u0027re not off by 1."},{"Start":"01:57.940 ","End":"01:59.360","Text":"We checked it."},{"Start":"01:59.360 ","End":"02:00.740","Text":"Just to remind you where we are,"},{"Start":"02:00.740 ","End":"02:08.219","Text":"we\u0027re going to let x equal Pi/2 here and here and here."},{"Start":"02:08.219 ","End":"02:12.050","Text":"Just copied it again with the x highlighted."},{"Start":"02:12.050 ","End":"02:16.570","Text":"We get Pi/2 times Pi minus Pi/2,"},{"Start":"02:16.570 ","End":"02:18.670","Text":"it\u0027s Pi^2 over 4."},{"Start":"02:18.670 ","End":"02:21.735","Text":"And here we have what this is,"},{"Start":"02:21.735 ","End":"02:24.880","Text":"and we have sin 2k minus 1 Pi/2,"},{"Start":"02:24.880 ","End":"02:33.480","Text":"which we\u0027ve just shown is equal to minus 1^k plus 1 over 2k minus 1^3."},{"Start":"02:33.480 ","End":"02:37.180","Text":"I also moved the Pi over to the other side so we get"},{"Start":"02:37.180 ","End":"02:42.890","Text":"Pi^3 and the 8 comes down to the bottom, Pi^3 over 32."},{"Start":"02:42.890 ","End":"02:44.750","Text":"If you go back and check,"},{"Start":"02:44.750 ","End":"02:47.270","Text":"this is exactly what we had to prove."},{"Start":"02:47.270 ","End":"02:50.165","Text":"Let\u0027s take a look. Yeah, there it is."},{"Start":"02:50.165 ","End":"02:53.520","Text":"We\u0027ve done it and that\u0027s Part D."}],"ID":28767},{"Watched":false,"Name":"Exercise 2 - Part e","Duration":"3m 28s","ChapterTopicVideoID":27604,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.400","Text":"We come to part e. This time we have"},{"Start":"00:05.400 ","End":"00:11.955","Text":"a cosine series as opposed to our original sine series."},{"Start":"00:11.955 ","End":"00:18.960","Text":"This works out because if we take this function and integrate it,"},{"Start":"00:18.960 ","End":"00:21.885","Text":"we get this function."},{"Start":"00:21.885 ","End":"00:30.645","Text":"We\u0027ll see that the term by term integration of this sine series gives this cosine series."},{"Start":"00:30.645 ","End":"00:36.230","Text":"We can show that this is the integral of this just by differentiating this,"},{"Start":"00:36.230 ","End":"00:38.015","Text":"and we get this."},{"Start":"00:38.015 ","End":"00:42.995","Text":"What we get is Pi x minus x^2,"},{"Start":"00:42.995 ","End":"00:45.440","Text":"and that\u0027s the same as this."},{"Start":"00:45.440 ","End":"00:49.895","Text":"G is the indefinite integral of f, or the antiderivative."},{"Start":"00:49.895 ","End":"00:52.955","Text":"Like I said, we\u0027ll use term by term integration."},{"Start":"00:52.955 ","End":"00:55.429","Text":"If you ask why we can do this,"},{"Start":"00:55.429 ","End":"00:59.045","Text":"to know we can always integrate a Fourier series term by term."},{"Start":"00:59.045 ","End":"01:02.330","Text":"The thing is you don\u0027t always get a Fourier series."},{"Start":"01:02.330 ","End":"01:05.060","Text":"If you have a non-zero constant term,"},{"Start":"01:05.060 ","End":"01:07.175","Text":"you\u0027ll get that times x,"},{"Start":"01:07.175 ","End":"01:09.110","Text":"which is not a sine or cosine."},{"Start":"01:09.110 ","End":"01:12.475","Text":"But in our case we will get a Fourier series."},{"Start":"01:12.475 ","End":"01:15.860","Text":"x times Pi minus x, we know,"},{"Start":"01:15.860 ","End":"01:19.955","Text":"is equal to this series on 0Pi."},{"Start":"01:19.955 ","End":"01:27.095","Text":"On 0Pi, g(x) will be the term by term integration of this,"},{"Start":"01:27.095 ","End":"01:29.524","Text":"which will give us this series."},{"Start":"01:29.524 ","End":"01:32.540","Text":"The sign comes out to be cosine,"},{"Start":"01:32.540 ","End":"01:34.960","Text":"and we divide by 2k minus 1,"},{"Start":"01:34.960 ","End":"01:37.750","Text":"which makes this 3 become a 4."},{"Start":"01:37.750 ","End":"01:41.105","Text":"We also have a minus, because the integral of sine is minus cosine."},{"Start":"01:41.105 ","End":"01:43.955","Text":"But the constant of integration, C,"},{"Start":"01:43.955 ","End":"01:47.600","Text":"which we\u0027ll call a_0 over 2 for convenience,"},{"Start":"01:47.600 ","End":"01:50.540","Text":"so we have a formula for a_0."},{"Start":"01:50.540 ","End":"01:54.215","Text":"Well usually the formula\u0027s 1/Pi,"},{"Start":"01:54.215 ","End":"01:56.525","Text":"and from minus Pi to Pi."},{"Start":"01:56.525 ","End":"02:00.860","Text":"But g, we can extend it to an even function,"},{"Start":"02:00.860 ","End":"02:03.590","Text":"G and all of minus Pi Pi."},{"Start":"02:03.590 ","End":"02:05.420","Text":"Then a_0 will be 1/Pi,"},{"Start":"02:05.420 ","End":"02:07.580","Text":"the integral minus Pi to Pi(G)."},{"Start":"02:07.580 ","End":"02:10.100","Text":"Then we can say, because it\u0027s even it\u0027s 2/Pi,"},{"Start":"02:10.100 ","End":"02:12.005","Text":"the integral from 0 to Pi."},{"Start":"02:12.005 ","End":"02:14.480","Text":"But on the interval from 0 to Pi,"},{"Start":"02:14.480 ","End":"02:19.430","Text":"G is the same as g. That\u0027s the logic of getting 2/Pi."},{"Start":"02:19.430 ","End":"02:22.220","Text":"Or you can remember that when it\u0027s an even function that"},{"Start":"02:22.220 ","End":"02:25.850","Text":"the a_0 coefficient is given always as 2/Pi,"},{"Start":"02:25.850 ","End":"02:27.865","Text":"the integral of the function."},{"Start":"02:27.865 ","End":"02:31.025","Text":"This is the integral we have to compute."},{"Start":"02:31.025 ","End":"02:33.335","Text":"G(x) is equal to this."},{"Start":"02:33.335 ","End":"02:36.905","Text":"Just a straightforward integral of a polynomial."},{"Start":"02:36.905 ","End":"02:39.650","Text":"x^2 becomes x^3 over 3,"},{"Start":"02:39.650 ","End":"02:41.945","Text":"and x^3 becomes x^4 over 4."},{"Start":"02:41.945 ","End":"02:44.390","Text":"Evaluate this from 0 to Pi."},{"Start":"02:44.390 ","End":"02:46.835","Text":"When we plug in 0, it\u0027s just 0."},{"Start":"02:46.835 ","End":"02:49.460","Text":"When we plug in Pi, we get a Pi^4 here."},{"Start":"02:49.460 ","End":"02:50.690","Text":"We get a Pi^4 here."},{"Start":"02:50.690 ","End":"02:52.730","Text":"Pull that Pi^4 in front,"},{"Start":"02:52.730 ","End":"02:57.220","Text":"we get 1/6 minus the 1/12, which is 1/12."},{"Start":"02:57.220 ","End":"03:00.810","Text":"We get Pi^4 over Pi is Pi^3,"},{"Start":"03:00.810 ","End":"03:04.320","Text":"and we have 2 times 1/12 is 1/6,"},{"Start":"03:04.320 ","End":"03:07.725","Text":"so a_0 is Pi^3 over 6."},{"Start":"03:07.725 ","End":"03:09.940","Text":"Now that we have a_0,"},{"Start":"03:09.940 ","End":"03:12.845","Text":"which was the final piece of the puzzle here,"},{"Start":"03:12.845 ","End":"03:15.520","Text":"we can plug it in and get,"},{"Start":"03:15.520 ","End":"03:18.735","Text":"this will be Pi^3 over 12."},{"Start":"03:18.735 ","End":"03:23.360","Text":"This is the formula for this function on 0Pi."},{"Start":"03:23.360 ","End":"03:28.260","Text":"That\u0027s what we needed for part e. We\u0027re done with this clip."}],"ID":28768},{"Watched":false,"Name":"Exercise 2 - Part f","Duration":"2m 17s","ChapterTopicVideoID":27605,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.825","Text":"Now we come to the last section, f,"},{"Start":"00:03.825 ","End":"00:06.480","Text":"where we\u0027re going to use the result of e,"},{"Start":"00:06.480 ","End":"00:09.660","Text":"the expansion of this as a cosine series,"},{"Start":"00:09.660 ","End":"00:19.125","Text":"to figure out that this series of 1 over odd numbers to the 4th adds up to Pi^4 over 96."},{"Start":"00:19.125 ","End":"00:24.870","Text":"This is like 1 over 1^4 plus 1 over 3^4 plus 1 over 5^4,"},{"Start":"00:24.870 ","End":"00:25.905","Text":"plus 1 over 7^4."},{"Start":"00:25.905 ","End":"00:29.175","Text":"Very interesting to know that that\u0027s Pi^4 over 96."},{"Start":"00:29.175 ","End":"00:35.700","Text":"Let\u0027s see what we had in Part e. We showed that if we call this function g(x),"},{"Start":"00:35.700 ","End":"00:40.725","Text":"then the cosine series for g(x),"},{"Start":"00:40.725 ","End":"00:44.150","Text":"this part is the cosine series."},{"Start":"00:44.150 ","End":"00:45.810","Text":"That\u0027s the a_0 over 2,"},{"Start":"00:45.810 ","End":"00:51.745","Text":"and this is the sum for the odd numbers of b_n cosine n_x."},{"Start":"00:51.745 ","End":"00:55.180","Text":"The hint says to substitute x=0,"},{"Start":"00:55.180 ","End":"00:57.770","Text":"presumably using Dirichlet theorem,"},{"Start":"00:57.770 ","End":"01:00.200","Text":"but 0 is an endpoint here."},{"Start":"01:00.200 ","End":"01:03.250","Text":"How about we extend it from minus Pi to Pi?"},{"Start":"01:03.250 ","End":"01:04.475","Text":"We\u0027ve already done that."},{"Start":"01:04.475 ","End":"01:09.380","Text":"We have the same series for minus Pi to Pi,"},{"Start":"01:09.380 ","End":"01:13.700","Text":"the big G, just that we consider it as a function from minus Pi to Pi."},{"Start":"01:13.700 ","End":"01:19.355","Text":"But 0 is an inside point and the function is continuous there."},{"Start":"01:19.355 ","End":"01:21.915","Text":"Because it\u0027s continuous, we can substitute,"},{"Start":"01:21.915 ","End":"01:24.165","Text":"instead of x, we can put 0."},{"Start":"01:24.165 ","End":"01:30.560","Text":"What we get is that Pi times 0^2 over 2 minus 0^3 over 3,"},{"Start":"01:30.560 ","End":"01:31.960","Text":"well, all this is 0,"},{"Start":"01:31.960 ","End":"01:36.560","Text":"is Pi^3 over 12 plus all of this with a 0 here."},{"Start":"01:36.560 ","End":"01:42.410","Text":"All these cosines are going to be equal to 1 because cosine of 0 is 1,"},{"Start":"01:42.410 ","End":"01:44.974","Text":"so this simplifies to this."},{"Start":"01:44.974 ","End":"01:51.890","Text":"Now we can bring the series to the other side and then multiply by Pi/8."},{"Start":"01:51.890 ","End":"01:53.955","Text":"Well, leave the series over here,"},{"Start":"01:53.955 ","End":"01:55.050","Text":"maybe we put this to the other side and then multiply by minus Pi/8."},{"Start":"01:57.460 ","End":"01:59.905","Text":"Anyway, we get this,"},{"Start":"01:59.905 ","End":"02:02.280","Text":"Pi^3 times Pi is Pi^4,"},{"Start":"02:02.280 ","End":"02:05.460","Text":"12 times 8 is 96."},{"Start":"02:05.460 ","End":"02:11.780","Text":"We get that this series sums to Pi^4 over 96."},{"Start":"02:11.780 ","End":"02:14.915","Text":"This is what we had to show."},{"Start":"02:14.915 ","End":"02:17.700","Text":"We are done."}],"ID":28769},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"2m 42s","ChapterTopicVideoID":27606,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"In this exercise, we consider"},{"Start":"00:03.000 ","End":"00:08.400","Text":"the function f(x) equals e to the x^2 on the interval from minus Pi to Pi."},{"Start":"00:08.400 ","End":"00:12.795","Text":"Let\u0027s say that it\u0027s complex Fourier expansion is this."},{"Start":"00:12.795 ","End":"00:17.100","Text":"Then we have 3 prove or disprove all relating to"},{"Start":"00:17.100 ","End":"00:22.649","Text":"series involving c_n and whether or not the series converges."},{"Start":"00:22.649 ","End":"00:24.720","Text":"We have to prove or disprove."},{"Start":"00:24.720 ","End":"00:27.660","Text":"In Part a, we want the series,"},{"Start":"00:27.660 ","End":"00:29.220","Text":"the sum of n^2,"},{"Start":"00:29.220 ","End":"00:31.410","Text":"absolute value of (c_n)^2."},{"Start":"00:31.410 ","End":"00:33.045","Text":"Let\u0027s do that."},{"Start":"00:33.045 ","End":"00:35.190","Text":"Turns out that it\u0027s true."},{"Start":"00:35.190 ","End":"00:36.720","Text":"We need to prove it."},{"Start":"00:36.720 ","End":"00:42.475","Text":"We\u0027ll use the following theorem that we had on term by term differentiation."},{"Start":"00:42.475 ","End":"00:46.490","Text":"If f satisfies that it\u0027s continuous on"},{"Start":"00:46.490 ","End":"00:52.205","Text":"the interval and has the same value at minus Pi and Pi, the endpoints,"},{"Start":"00:52.205 ","End":"00:57.355","Text":"and if it\u0027s piecewise continuously differentiable on the interval,"},{"Start":"00:57.355 ","End":"01:00.830","Text":"then we can differentiate the series term by term,"},{"Start":"01:00.830 ","End":"01:03.125","Text":"which means that f\u0027,"},{"Start":"01:03.125 ","End":"01:04.700","Text":"the derivative of this,"},{"Start":"01:04.700 ","End":"01:07.850","Text":"is the sum inc_n to the inx."},{"Start":"01:07.850 ","End":"01:10.965","Text":"We just get an extra factor of in here."},{"Start":"01:10.965 ","End":"01:12.825","Text":"Now, our function, f,"},{"Start":"01:12.825 ","End":"01:15.970","Text":"satisfies the conditions of the theorem."},{"Start":"01:15.970 ","End":"01:18.290","Text":"Here\u0027s a reminder of what f is."},{"Start":"01:18.290 ","End":"01:20.435","Text":"Well, it\u0027s continuous."},{"Start":"01:20.435 ","End":"01:22.010","Text":"It\u0027s also differentiable."},{"Start":"01:22.010 ","End":"01:24.500","Text":"The derivative is also continuous."},{"Start":"01:24.500 ","End":"01:26.420","Text":"We don\u0027t need even the piecewise."},{"Start":"01:26.420 ","End":"01:30.850","Text":"Now, it\u0027s equal at the endpoints because if you plug in Pi or minus Pi,"},{"Start":"01:30.850 ","End":"01:32.720","Text":"it\u0027s the same, it\u0027s an even function."},{"Start":"01:32.720 ","End":"01:36.650","Text":"Now, we don\u0027t need anymore of what f actually is,"},{"Start":"01:36.650 ","End":"01:40.090","Text":"just that it satisfies the conditions."},{"Start":"01:40.090 ","End":"01:45.455","Text":"The derivative is equal to the sum of inc_n e to the inx."},{"Start":"01:45.455 ","End":"01:49.670","Text":"Now, we can apply Parseval\u0027s theorem not to f but to f\u0027."},{"Start":"01:49.670 ","End":"01:55.670","Text":"We get that the sum of the absolute value squared of just the coefficient,"},{"Start":"01:55.670 ","End":"02:01.955","Text":"the inc_n, it converges to 1 over 2Pi the integral of the function squared."},{"Start":"02:01.955 ","End":"02:08.285","Text":"The function is f\u0027. This is finite because f\u0027 is continuous,"},{"Start":"02:08.285 ","End":"02:11.766","Text":"and therefore, bounded on a finite interval,"},{"Start":"02:11.766 ","End":"02:13.745","Text":"so the integral is going to be finite."},{"Start":"02:13.745 ","End":"02:18.335","Text":"Just to remind you, this is what we had to prove that it converges,"},{"Start":"02:18.335 ","End":"02:20.930","Text":"which is not quite the same as this."},{"Start":"02:20.930 ","End":"02:26.190","Text":"But the absolute value of i is 1, get rid of that."},{"Start":"02:26.190 ","End":"02:30.585","Text":"Also, absolute value of n^2, just n^2."},{"Start":"02:30.585 ","End":"02:32.580","Text":"This is equal to this."},{"Start":"02:32.580 ","End":"02:36.800","Text":"This series is the same as the series that we had."},{"Start":"02:36.800 ","End":"02:38.150","Text":"This converges."},{"Start":"02:38.150 ","End":"02:39.860","Text":"That\u0027s what we wanted."},{"Start":"02:39.860 ","End":"02:42.780","Text":"That concludes Part A."}],"ID":28770},{"Watched":false,"Name":"Exercise 3 - Part b","Duration":"1m 20s","ChapterTopicVideoID":27607,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.130","Text":"Now we come to part b."},{"Start":"00:02.130 ","End":"00:04.140","Text":"We\u0027ve just done part a, and in fact,"},{"Start":"00:04.140 ","End":"00:07.020","Text":"we\u0027ll use part a to prove part b."},{"Start":"00:07.020 ","End":"00:12.440","Text":"We have to show that the series of sum of absolute value of c_n converges."},{"Start":"00:12.440 ","End":"00:14.715","Text":"Like I said, we\u0027ll prove it using part a."},{"Start":"00:14.715 ","End":"00:19.845","Text":"We have this converges and we\u0027re going to use a trick that we\u0027ve used before."},{"Start":"00:19.845 ","End":"00:24.060","Text":"We can write c_n as 1 over n times n times c_n,"},{"Start":"00:24.060 ","End":"00:26.550","Text":"but we have to exclude n=0,"},{"Start":"00:26.550 ","End":"00:28.455","Text":"we\u0027ll take care of that later."},{"Start":"00:28.455 ","End":"00:32.460","Text":"Then we can use the standard inequality that"},{"Start":"00:32.460 ","End":"00:37.200","Text":"absolute value of xy is less than or equal to 1/2 of x^2 plus y^2."},{"Start":"00:37.200 ","End":"00:43.490","Text":"If we use it, this is x and this is y and we get this is less than or equal to this."},{"Start":"00:43.490 ","End":"00:48.710","Text":"Then we can break this sum up into 2 sums because each of them converges."},{"Start":"00:48.710 ","End":"00:53.450","Text":"The sum of 1 over n^2 is known to converge to Pi^2 over 6,"},{"Start":"00:53.450 ","End":"00:58.380","Text":"and this 1 converges because we had it in part a."},{"Start":"00:58.380 ","End":"01:02.615","Text":"Sum of 2 convergent series is convergent and this is what it\u0027s equal."},{"Start":"01:02.615 ","End":"01:06.020","Text":"Just as a matter of the n not equal to 0,"},{"Start":"01:06.020 ","End":"01:07.460","Text":"that\u0027s no big deal."},{"Start":"01:07.460 ","End":"01:09.230","Text":"It\u0027s just 1 term."},{"Start":"01:09.230 ","End":"01:12.230","Text":"If this is finite convergent,"},{"Start":"01:12.230 ","End":"01:13.460","Text":"then so is this."},{"Start":"01:13.460 ","End":"01:16.430","Text":"Just adding a finite number doesn\u0027t change anything,"},{"Start":"01:16.430 ","End":"01:21.390","Text":"and that proves part b and we are done."}],"ID":28771},{"Watched":false,"Name":"Exercise 3 - Part c","Duration":"4m 28s","ChapterTopicVideoID":27608,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"Now we just proved Parts a and b,"},{"Start":"00:03.480 ","End":"00:05.445","Text":"Part c turns out to be false,"},{"Start":"00:05.445 ","End":"00:07.965","Text":"and we\u0027re going to disprove it."},{"Start":"00:07.965 ","End":"00:12.135","Text":"Remember that f(x) is e to the x^2."},{"Start":"00:12.135 ","End":"00:15.285","Text":"F\u0027(x) is 2xe to the x^2."},{"Start":"00:15.285 ","End":"00:20.220","Text":"Note that if you substitute the endpoints minus Pi and Pi,"},{"Start":"00:20.220 ","End":"00:21.900","Text":"you get different things."},{"Start":"00:21.900 ","End":"00:27.255","Text":"What this means is that if we extend f\u0027 to a periodic function,"},{"Start":"00:27.255 ","End":"00:29.720","Text":"an R, give it a name,"},{"Start":"00:29.720 ","End":"00:31.745","Text":"f is not continuous."},{"Start":"00:31.745 ","End":"00:34.595","Text":"Let\u0027s just illustrate that with the graph."},{"Start":"00:34.595 ","End":"00:37.625","Text":"At Pi, it\u0027s equal to,"},{"Start":"00:37.625 ","End":"00:39.080","Text":"well, some large number,"},{"Start":"00:39.080 ","End":"00:44.555","Text":"but positive it\u0027s equal to this number and at minus Pi it\u0027s equal to minus that number."},{"Start":"00:44.555 ","End":"00:47.000","Text":"In fact, the derivative is an odd function."},{"Start":"00:47.000 ","End":"00:49.265","Text":"If we want to extend it periodically,"},{"Start":"00:49.265 ","End":"00:52.495","Text":"we\u0027ll have these jumps at Pi,"},{"Start":"00:52.495 ","End":"00:53.930","Text":"at 3Pi and so on."},{"Start":"00:53.930 ","End":"00:59.825","Text":"We just can\u0027t get it to be continuous if we also want it to be periodic, doesn\u0027t work."},{"Start":"00:59.825 ","End":"01:02.760","Text":"Anyway. Bear this in mind."},{"Start":"01:02.760 ","End":"01:04.870","Text":"Recall from Part A,"},{"Start":"01:04.870 ","End":"01:08.825","Text":"we showed that f prime is this series."},{"Start":"01:08.825 ","End":"01:12.290","Text":"Basically, we did a term by term differentiation."},{"Start":"01:12.290 ","End":"01:16.070","Text":"Now we\u0027ll prove our claim by contradiction."},{"Start":"01:16.070 ","End":"01:22.205","Text":"Suppose on the contrary that this does converge and we\u0027ll reach a contradiction."},{"Start":"01:22.205 ","End":"01:25.700","Text":"The plan is, give you the outline."},{"Start":"01:25.700 ","End":"01:31.399","Text":"First thing is we\u0027ll use the Weierstrass M-test to show that this series converges"},{"Start":"01:31.399 ","End":"01:37.370","Text":"uniformly on R. Then it follows that the limit is F,"},{"Start":"01:37.370 ","End":"01:40.160","Text":"which is the periodic extension of f\u0027."},{"Start":"01:40.160 ","End":"01:43.455","Text":"This will imply that F(x),"},{"Start":"01:43.455 ","End":"01:46.040","Text":"which is the sum of"},{"Start":"01:46.040 ","End":"01:50.860","Text":"a uniformly convergent series of continuous functions is also continuous,"},{"Start":"01:50.860 ","End":"01:54.690","Text":"and we\u0027ll use this to show it."},{"Start":"01:54.690 ","End":"02:01.189","Text":"Then we\u0027ll get a contradiction because we saw that F(x) is not continuous."},{"Start":"02:01.189 ","End":"02:03.335","Text":"It jumps at Pi, for example."},{"Start":"02:03.335 ","End":"02:07.370","Text":"Let\u0027s remember what the Weierstrass M-test is."},{"Start":"02:07.370 ","End":"02:14.315","Text":"Suppose that f_n is a sequence of complex valued functions defined on a set,"},{"Start":"02:14.315 ","End":"02:20.240","Text":"and also suppose that there is a sequence of positive real numbers M_n,"},{"Start":"02:20.240 ","End":"02:26.235","Text":"such that the absolute value of f_n(x) less than or equal to M_n for all n,"},{"Start":"02:26.235 ","End":"02:29.775","Text":"and importantly for all x in the set."},{"Start":"02:29.775 ","End":"02:34.565","Text":"We suppose that the series sum of M_n is convergent."},{"Start":"02:34.565 ","End":"02:37.160","Text":"Then we conclude that the series,"},{"Start":"02:37.160 ","End":"02:43.420","Text":"the sum of f_n(x) converges uniformly on a little so absolutely."},{"Start":"02:43.420 ","End":"02:49.505","Text":"We take f_n(x) to be the general term in our series here."},{"Start":"02:49.505 ","End":"02:56.615","Text":"So this will be f_n(x) and it satisfies the conditions of the Weierstrass M-test,"},{"Start":"02:56.615 ","End":"03:01.760","Text":"if we take M_n to be n times absolute value of C_n,"},{"Start":"03:01.760 ","End":"03:08.910","Text":"because we know that the absolute value of f_n from here is equal to n. The i,"},{"Start":"03:08.910 ","End":"03:10.440","Text":"has absolute value 1,"},{"Start":"03:10.440 ","End":"03:12.515","Text":"and e to the inx has absolute value 1."},{"Start":"03:12.515 ","End":"03:14.975","Text":"We just get this, we call this M_n,"},{"Start":"03:14.975 ","End":"03:17.600","Text":"then it satisfies the conditions."},{"Start":"03:17.600 ","End":"03:19.355","Text":"The sum of M_n,"},{"Start":"03:19.355 ","End":"03:21.020","Text":"which is the sum of this,"},{"Start":"03:21.020 ","End":"03:24.350","Text":"is finite series converges."},{"Start":"03:24.350 ","End":"03:27.080","Text":"This is our assumption in the proof by contradiction,"},{"Start":"03:27.080 ","End":"03:30.410","Text":"we\u0027re assuming that this is convergent."},{"Start":"03:30.410 ","End":"03:32.390","Text":"By the M test,"},{"Start":"03:32.390 ","End":"03:36.770","Text":"series sum of f_n of x converges uniformly."},{"Start":"03:36.770 ","End":"03:40.250","Text":"The theorem just says that the series converges,"},{"Start":"03:40.250 ","End":"03:41.930","Text":"but we know what its limit is."},{"Start":"03:41.930 ","End":"03:43.340","Text":"If it converges uniformly,"},{"Start":"03:43.340 ","End":"03:47.465","Text":"it has to be convergent to the pointwise limit, which is F(x)."},{"Start":"03:47.465 ","End":"03:51.395","Text":"Now we\u0027re going to use another theorem on uniform convergence."},{"Start":"03:51.395 ","End":"03:55.610","Text":"If we have a series of continuous functions that converges"},{"Start":"03:55.610 ","End":"04:01.040","Text":"uniformly then the sum is also continuous on our domain."},{"Start":"04:01.040 ","End":"04:05.690","Text":"F is continuous by this theorem,"},{"Start":"04:05.690 ","End":"04:12.545","Text":"and that\u0027s a contradiction because what we had to remember is that f is not continuous."},{"Start":"04:12.545 ","End":"04:16.114","Text":"Remember it jumps at x equals Pi, for example."},{"Start":"04:16.114 ","End":"04:20.255","Text":"So this contradiction means that this assumption can\u0027t be true,"},{"Start":"04:20.255 ","End":"04:23.360","Text":"and so this series is not convergent."},{"Start":"04:23.360 ","End":"04:25.115","Text":"That concludes Part C,"},{"Start":"04:25.115 ","End":"04:28.409","Text":"and that\u0027s the last part in this exercise."}],"ID":28772},{"Watched":false,"Name":"Exercise 4","Duration":"9m 9s","ChapterTopicVideoID":27609,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.390","Text":"In this exercise we\u0027re going to consider the following series of complex functions,"},{"Start":"00:06.390 ","End":"00:09.255","Text":"the sum of e^in x plus I."},{"Start":"00:09.255 ","End":"00:12.240","Text":"We have to show that this series of functions"},{"Start":"00:12.240 ","End":"00:16.650","Text":"converges uniformly to something and we call that something f(x)."},{"Start":"00:16.650 ","End":"00:18.645","Text":"In Part b the question is,"},{"Start":"00:18.645 ","End":"00:24.855","Text":"how many times can we differentiate term-by-term this series?"},{"Start":"00:24.855 ","End":"00:28.215","Text":"To proving the uniform convergence,"},{"Start":"00:28.215 ","End":"00:30.960","Text":"we often use the Weierstrass M-test,"},{"Start":"00:30.960 ","End":"00:32.430","Text":"and that\u0027s what we\u0027ll do here."},{"Start":"00:32.430 ","End":"00:33.750","Text":"I\u0027ll remind you what it is,"},{"Start":"00:33.750 ","End":"00:36.510","Text":"I copied this version from the Wikipedia."},{"Start":"00:36.510 ","End":"00:39.425","Text":"We have a sequence of functions f_n."},{"Start":"00:39.425 ","End":"00:43.280","Text":"Suppose that they have the condition that they\u0027re bounded,"},{"Start":"00:43.280 ","End":"00:51.310","Text":"for each n the function f_n(x) is bounded on all its domain A by a constant M_n."},{"Start":"00:51.310 ","End":"00:55.460","Text":"Actually, it should just be for all n. Whatever n that we\u0027re summing on,"},{"Start":"00:55.460 ","End":"01:00.955","Text":"in this case it\u0027ll be for all n from 0 to infinity: 0, 1, 2, etc."},{"Start":"01:00.955 ","End":"01:06.270","Text":"Again, each f_n is bounded by a constant M_n."},{"Start":"01:06.270 ","End":"01:11.465","Text":"We\u0027re also given that the sum of these constants M_n converges."},{"Start":"01:11.465 ","End":"01:14.363","Text":"If these two conditions are met then the series;"},{"Start":"01:14.363 ","End":"01:18.590","Text":"the sum of f_n of x converges uniformly we care about,"},{"Start":"01:18.590 ","End":"01:21.950","Text":"but it\u0027s also absolutely on the domain A."},{"Start":"01:21.950 ","End":"01:26.165","Text":"Let\u0027s prove what we want to prove using Weierstrass M test."},{"Start":"01:26.165 ","End":"01:29.710","Text":"Let\u0027s call this the general term f_n(x)."},{"Start":"01:29.710 ","End":"01:32.359","Text":"It\u0027s the general term that\u0027s in the series."},{"Start":"01:32.359 ","End":"01:35.725","Text":"This is what we\u0027ll call f_n(x),"},{"Start":"01:35.725 ","End":"01:39.585","Text":"and this is what it\u0027s equal to but we can rewrite it."},{"Start":"01:39.585 ","End":"01:42.950","Text":"Because i times i is minus 1,"},{"Start":"01:42.950 ","End":"01:46.900","Text":"this will come out to be e^minus n, e^inx."},{"Start":"01:46.900 ","End":"01:52.070","Text":"Actually, our series is a Fourier series complex."},{"Start":"01:52.070 ","End":"01:56.630","Text":"But we\u0027re just considering it as a general function series."},{"Start":"01:56.630 ","End":"02:00.175","Text":"Now let\u0027s estimate the absolute value of each f_n."},{"Start":"02:00.175 ","End":"02:05.255","Text":"Absolute value of f_n(x) is the absolute value of e^minus n, e^inx."},{"Start":"02:05.255 ","End":"02:09.610","Text":"Now the absolute value of e^i something is 1."},{"Start":"02:09.610 ","End":"02:12.365","Text":"It\u0027s always on the unit circle."},{"Start":"02:12.365 ","End":"02:16.340","Text":"The absolute value of e^minus n which is positive is just e^minus"},{"Start":"02:16.340 ","End":"02:22.440","Text":"n. We\u0027ll define this to be M_n for the theorem here."},{"Start":"02:22.440 ","End":"02:27.045","Text":"The sum of the series of constants M_n is the sum of e^minus n"},{"Start":"02:27.045 ","End":"02:32.480","Text":"and it\u0027s easy to prove that it converges generally using this root test,"},{"Start":"02:32.480 ","End":"02:37.138","Text":"but here we actually can say what the limit is because it\u0027s a geometric series."},{"Start":"02:37.138 ","End":"02:42.360","Text":"I\u0027m using the formula for the geometric series with q=e^minus 1."},{"Start":"02:42.360 ","End":"02:47.268","Text":"We have e^minus 1 over 1 minus e^minus 1,"},{"Start":"02:47.268 ","End":"02:49.070","Text":"multiply top and bottom by e. Anyway,"},{"Start":"02:49.070 ","End":"02:51.020","Text":"this is what we get, but that\u0027s not important."},{"Start":"02:51.020 ","End":"02:52.580","Text":"The thing is that it\u0027s finite."},{"Start":"02:52.580 ","End":"02:56.210","Text":"We\u0027ve met the conditions both of them for the M tests."},{"Start":"02:56.210 ","End":"02:59.840","Text":"We can get the conclusion that the sum f_n(x),"},{"Start":"02:59.840 ","End":"03:01.128","Text":"which is the sum that we want,"},{"Start":"03:01.128 ","End":"03:06.275","Text":"converges uniformly [inaudible] and convergences to something,"},{"Start":"03:06.275 ","End":"03:08.060","Text":"and we\u0027ll label that something f(x)."},{"Start":"03:08.060 ","End":"03:10.465","Text":"That\u0027s Part a."},{"Start":"03:10.465 ","End":"03:12.680","Text":"Now Part b."},{"Start":"03:12.680 ","End":"03:17.381","Text":"For Part b we\u0027ll need a theorem on term-by-term differentiation."},{"Start":"03:17.381 ","End":"03:20.540","Text":"The general series of function is not necessarily Fourier,"},{"Start":"03:20.540 ","End":"03:22.594","Text":"although in our case it is Fourier."},{"Start":"03:22.594 ","End":"03:29.756","Text":"Suppose u_n is a sequence of differentiable functions on some interval a, b."},{"Start":"03:29.756 ","End":"03:32.690","Text":"Suppose that both the sum of u_n,"},{"Start":"03:32.690 ","End":"03:38.105","Text":"and the sum of u_n\u0027 converge uniformly on this interval,"},{"Start":"03:38.105 ","End":"03:41.420","Text":"then the sum of u_n is"},{"Start":"03:41.420 ","End":"03:46.520","Text":"differentiable and the derivative of the sum of u_n is the sum of the derivatives."},{"Start":"03:46.520 ","End":"03:50.090","Text":"You can call it u_n\u0027(x) or d by dx u_n."},{"Start":"03:50.090 ","End":"03:56.460","Text":"Now what we\u0027re going to do with this theorem is we\u0027ll use it to show that for any k,"},{"Start":"03:56.460 ","End":"03:57.770","Text":"it could be 0 or 1,"},{"Start":"03:57.770 ","End":"03:58.970","Text":"2, 3, etc,"},{"Start":"03:58.970 ","End":"04:05.525","Text":"the kth derivative of f is the sum of the kth derivatives of the f_n."},{"Start":"04:05.525 ","End":"04:09.395","Text":"What this means if we rephrase it is that the series f"},{"Start":"04:09.395 ","End":"04:13.700","Text":"which is the sum of f_n can be differentiated infinitely many times."},{"Start":"04:13.700 ","End":"04:16.790","Text":"If k is 0, it\u0027s just the original series."},{"Start":"04:16.790 ","End":"04:21.845","Text":"If k is 1 it\u0027s the first derivative, and so on."},{"Start":"04:21.845 ","End":"04:25.760","Text":"Now recall that a function f_n(x),"},{"Start":"04:25.760 ","End":"04:29.925","Text":"we could rewrite it as e^minus n e^inx."},{"Start":"04:29.925 ","End":"04:31.835","Text":"Now, using this,"},{"Start":"04:31.835 ","End":"04:34.472","Text":"we can differentiate any number of times."},{"Start":"04:34.472 ","End":"04:35.910","Text":"What will happen,"},{"Start":"04:35.910 ","End":"04:38.177","Text":"this is a constant as far as x goes."},{"Start":"04:38.177 ","End":"04:42.390","Text":"Here\u0027s e^inx, we\u0027ll multiply by in."},{"Start":"04:42.390 ","End":"04:45.540","Text":"Each time we differentiate then the in will be a constant."},{"Start":"04:45.540 ","End":"04:49.685","Text":"We\u0027ll get in to the power of however many times we differentiate."},{"Start":"04:49.685 ","End":"04:52.970","Text":"This works for k equals 0 also by the way."},{"Start":"04:52.970 ","End":"04:54.650","Text":"I think this is fairly clear though."},{"Start":"04:54.650 ","End":"04:55.760","Text":"If we wanted to be precise,"},{"Start":"04:55.760 ","End":"04:57.545","Text":"we do it by induction."},{"Start":"04:57.545 ","End":"05:05.690","Text":"Each time we\u0027d increase k by 1 and we\u0027d get an extra in thrown into this exponent,"},{"Start":"05:05.690 ","End":"05:07.535","Text":"so we\u0027d get k plus 1."},{"Start":"05:07.535 ","End":"05:11.570","Text":"The next thing we\u0027re going to do is show that the sum of"},{"Start":"05:11.570 ","End":"05:15.950","Text":"these f_n^k sum on n converges uniformly."},{"Start":"05:15.950 ","End":"05:18.605","Text":"Again, we\u0027re going to use the Weierstrass M test."},{"Start":"05:18.605 ","End":"05:22.340","Text":"First, let\u0027s evaluate each general term and absolute value."},{"Start":"05:22.340 ","End":"05:25.850","Text":"This is equal to the absolute value of this which you just got from here."},{"Start":"05:25.850 ","End":"05:29.150","Text":"Each of the inx has absolute value 1."},{"Start":"05:29.150 ","End":"05:31.790","Text":"I also has absolute value 1."},{"Start":"05:31.790 ","End":"05:33.845","Text":"We just get n^k,"},{"Start":"05:33.845 ","End":"05:39.295","Text":"e^minus n. Put that as e^n in the denominator and we\u0027ll call this m_n."},{"Start":"05:39.295 ","End":"05:45.195","Text":"The sum of the series m_n is just the sum of the series n^k over e^n,"},{"Start":"05:45.195 ","End":"05:49.845","Text":"and this converges using the root test."},{"Start":"05:49.845 ","End":"05:54.355","Text":"The nth root of the nth term is,"},{"Start":"05:54.355 ","End":"06:01.675","Text":"you can just take the nth root of n^k and the nth root of e^n is e. Famous limit."},{"Start":"06:01.675 ","End":"06:08.250","Text":"The limit of nth root of n as n goes to infinity this part comes out to be 1."},{"Start":"06:08.250 ","End":"06:09.490","Text":"The answer is 1 over e,"},{"Start":"06:09.490 ","End":"06:11.410","Text":"and the point is that it\u0027s less than 1."},{"Start":"06:11.410 ","End":"06:15.820","Text":"The root test says that when this limit exists and is less than 1,"},{"Start":"06:15.820 ","End":"06:18.815","Text":"then the series converges."},{"Start":"06:18.815 ","End":"06:25.415","Text":"The M-test shows us that this series converges uniformly."},{"Start":"06:25.415 ","End":"06:30.475","Text":"We\u0027ve used both the root test to show that the sum of m_n"},{"Start":"06:30.475 ","End":"06:36.140","Text":"converges and then the Weierstrass M-test to show that this converges uniformly."},{"Start":"06:36.140 ","End":"06:38.090","Text":"We claim that this series,"},{"Start":"06:38.090 ","End":"06:40.130","Text":"which we know converges uniformly,"},{"Start":"06:40.130 ","End":"06:44.690","Text":"specifically converges to the kth derivative of f,"},{"Start":"06:44.690 ","End":"06:49.160","Text":"and we\u0027ll prove it by induction on k. First of all,"},{"Start":"06:49.160 ","End":"06:50.900","Text":"take the case k equals 0."},{"Start":"06:50.900 ","End":"06:52.100","Text":"We\u0027ve actually proved this."},{"Start":"06:52.100 ","End":"06:53.705","Text":"This was Part a."},{"Start":"06:53.705 ","End":"06:55.490","Text":"Okay, I just jump back."},{"Start":"06:55.490 ","End":"07:01.670","Text":"This is where it says that the sum of f_n is f. That\u0027s the case k equals 0."},{"Start":"07:01.670 ","End":"07:05.310","Text":"Now we want the induction step to go from k- k plus 1."},{"Start":"07:05.310 ","End":"07:08.030","Text":"We\u0027ll assume we\u0027ve proved it for k and that"},{"Start":"07:08.030 ","End":"07:11.270","Text":"this holds for a particular k. We have to show"},{"Start":"07:11.270 ","End":"07:18.120","Text":"that the k plus 1s derivative is equal to the sum with k plus 1 here,"},{"Start":"07:18.120 ","End":"07:19.750","Text":"but we actually have to show that this is"},{"Start":"07:19.750 ","End":"07:24.040","Text":"differentiable and that it\u0027s derivative is this."},{"Start":"07:24.040 ","End":"07:26.800","Text":"For that we\u0027ll use the theorem we had."},{"Start":"07:26.800 ","End":"07:29.215","Text":"This is the theorem I\u0027m talking about."},{"Start":"07:29.215 ","End":"07:31.284","Text":"Now, k is fixed."},{"Start":"07:31.284 ","End":"07:35.950","Text":"Let u_n(x) be the function f_n^k(x),"},{"Start":"07:35.950 ","End":"07:40.360","Text":"the kth derivative of the function f_n so that"},{"Start":"07:40.360 ","End":"07:46.480","Text":"the derivative of u_n is the k plus 1nth derivative."},{"Start":"07:46.480 ","End":"07:50.470","Text":"Now we just showed that this series converges"},{"Start":"07:50.470 ","End":"07:55.934","Text":"uniformly for any k. In particular for our k and also for k plus 1."},{"Start":"07:55.934 ","End":"07:59.385","Text":"Because k could be anything here."},{"Start":"07:59.385 ","End":"08:06.050","Text":"That means that u_n(x) converges uniformly and so does u_n\u0027."},{"Start":"08:06.050 ","End":"08:13.358","Text":"Now we can apply the theorem which says that this sum is differentiable,"},{"Start":"08:13.358 ","End":"08:18.575","Text":"moreover it\u0027s derivative is gotten by term-by-term differentiation."},{"Start":"08:18.575 ","End":"08:20.000","Text":"Let\u0027s interpret this."},{"Start":"08:20.000 ","End":"08:26.517","Text":"U_n is f_n^k(x) both here and here."},{"Start":"08:26.517 ","End":"08:32.195","Text":"Now this derivative with respect to x of the kth derivative is the k plus 1nth,"},{"Start":"08:32.195 ","End":"08:35.080","Text":"k plus 1st derivative."},{"Start":"08:35.080 ","End":"08:41.730","Text":"We know this is f^k(x),"},{"Start":"08:41.730 ","End":"08:44.885","Text":"so we have the derivative of that is, this."},{"Start":"08:44.885 ","End":"08:48.200","Text":"Then we get what we wanted because the derivative of"},{"Start":"08:48.200 ","End":"08:52.367","Text":"the kth derivative again is the k plus 1st derivative."},{"Start":"08:52.367 ","End":"08:54.605","Text":"The right-hand side just leave it as is."},{"Start":"08:54.605 ","End":"08:57.785","Text":"We have k plus 1 and k plus 1,"},{"Start":"08:57.785 ","End":"09:02.695","Text":"and we started with k and k. That\u0027s the inductive step."},{"Start":"09:02.695 ","End":"09:09.420","Text":"This was the only missing step that we had to complete the proof. We are done."}],"ID":28773},{"Watched":false,"Name":"Exercise 5 - Part a","Duration":"9m 13s","ChapterTopicVideoID":27610,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"This exercise has 2 parts."},{"Start":"00:02.400 ","End":"00:06.435","Text":"In the first part, we\u0027re going to show that this Fourier series,"},{"Start":"00:06.435 ","End":"00:14.760","Text":"the sum of sine nx over n converges to Pi minus x over 2."},{"Start":"00:14.760 ","End":"00:18.524","Text":"It\u0027s not just that this is the Fourier series for this,"},{"Start":"00:18.524 ","End":"00:20.700","Text":"there\u0027s an actual equality here,"},{"Start":"00:20.700 ","End":"00:24.330","Text":"at least for x between 0 and 2 Pi."},{"Start":"00:24.330 ","End":"00:26.610","Text":"What happens at the endpoints?"},{"Start":"00:26.610 ","End":"00:29.370","Text":"That\u0027s what we\u0027ll show during the solution of Part A."},{"Start":"00:29.370 ","End":"00:31.860","Text":"In Part B, we\u0027ll read it when we come to it,"},{"Start":"00:31.860 ","End":"00:34.335","Text":"and meanwhile we\u0027ll start on the solution for A."},{"Start":"00:34.335 ","End":"00:42.140","Text":"Let\u0027s give this function a name f(x) is Pi minus x over 2 on the interval from 0 to 2 Pi."},{"Start":"00:42.140 ","End":"00:45.455","Text":"Let\u0027s find the real Fourier series for f,"},{"Start":"00:45.455 ","End":"00:49.310","Text":"and it will turn out that the real Fourier series is this,"},{"Start":"00:49.310 ","End":"00:51.680","Text":"then we\u0027ll discuss the equality."},{"Start":"00:51.680 ","End":"00:56.210","Text":"So f(x) is represented by a_0 over 2 plus,"},{"Start":"00:56.210 ","End":"00:58.235","Text":"well, the usual sum."},{"Start":"00:58.235 ","End":"01:03.230","Text":"The formula for the coefficients a_n is the standard one,"},{"Start":"01:03.230 ","End":"01:06.000","Text":"this, and it works for a_0 also."},{"Start":"01:06.000 ","End":"01:08.165","Text":"Let\u0027s do the computation."},{"Start":"01:08.165 ","End":"01:10.060","Text":"We know what f(x) is,"},{"Start":"01:10.060 ","End":"01:17.711","Text":"it\u0027s Pi-x/2, and the integral from 0 to 2 Pi. This is what we get."},{"Start":"01:17.711 ","End":"01:19.625","Text":"Now, let\u0027s make a substitution."},{"Start":"01:19.625 ","End":"01:23.450","Text":"We\u0027ll, let t be Pi minus x,"},{"Start":"01:23.450 ","End":"01:27.250","Text":"and therefore, x will be Pi minus t,"},{"Start":"01:27.250 ","End":"01:30.589","Text":"dt will be minus dx,"},{"Start":"01:30.589 ","End":"01:32.825","Text":"just the derivative of this,"},{"Start":"01:32.825 ","End":"01:42.825","Text":"and limits of integration will move from 0 to Pi to Pi minus 0,"},{"Start":"01:42.825 ","End":"01:45.060","Text":"Pi minus 2 Pi,"},{"Start":"01:45.060 ","End":"01:47.150","Text":"so it\u0027s from Pi to minus Pi."},{"Start":"01:47.150 ","End":"01:49.745","Text":"The upper limit is smaller than the lower limit,"},{"Start":"01:49.745 ","End":"01:53.750","Text":"so we\u0027ll switch these 2 around and throw out the minus here,"},{"Start":"01:53.750 ","End":"01:55.884","Text":"and that will be okay."},{"Start":"01:55.884 ","End":"01:58.760","Text":"We get 1/Pi,"},{"Start":"01:58.760 ","End":"02:04.880","Text":"Pi-x is t/2 cosine of then Pi minus nt."},{"Start":"02:04.880 ","End":"02:10.540","Text":"We\u0027ll use the trigonometric identity for cosine of a difference,"},{"Start":"02:10.540 ","End":"02:15.705","Text":"n Pi will be Alpha and nt will be Beta."},{"Start":"02:15.705 ","End":"02:19.150","Text":"We\u0027ve got cosine Alpha,"},{"Start":"02:19.150 ","End":"02:24.215","Text":"cosine Beta plus sine Alpha, sine Beta."},{"Start":"02:24.215 ","End":"02:26.070","Text":"Cosine n Pi,"},{"Start":"02:26.070 ","End":"02:29.370","Text":"we know this is minus 1^n."},{"Start":"02:29.370 ","End":"02:31.980","Text":"Sine n Pi is 0,"},{"Start":"02:31.980 ","End":"02:36.404","Text":"so what we\u0027re left with the t is here."},{"Start":"02:36.404 ","End":"02:39.540","Text":"The 2 goes in front of the integral,"},{"Start":"02:39.540 ","End":"02:41.520","Text":"so it\u0027s the 2 Pi,"},{"Start":"02:41.520 ","End":"02:46.680","Text":"also the minus 1^n can be pulled in front of the integral."},{"Start":"02:46.680 ","End":"02:51.705","Text":"What we\u0027re left with then is t cosine nt dt."},{"Start":"02:51.705 ","End":"02:59.380","Text":"Now, t is an odd function of t on this symmetric interval and cosine of nt is even."},{"Start":"02:59.380 ","End":"03:01.890","Text":"Odd times even is odd,"},{"Start":"03:01.890 ","End":"03:07.235","Text":"so we have the integral of an odd function on a symmetric interval,"},{"Start":"03:07.235 ","End":"03:10.265","Text":"and so this comes out to be 0."},{"Start":"03:10.265 ","End":"03:12.140","Text":"This works for all the n,"},{"Start":"03:12.140 ","End":"03:15.095","Text":"doesn\u0027t matter if n is 0."},{"Start":"03:15.095 ","End":"03:16.805","Text":"So all the a_n\u0027s,"},{"Start":"03:16.805 ","End":"03:18.433","Text":"including a_0 are 0\u0027s,"},{"Start":"03:18.433 ","End":"03:20.000","Text":"so we just have b_n\u0027s."},{"Start":"03:20.000 ","End":"03:22.555","Text":"Let\u0027s compute what b_n is."},{"Start":"03:22.555 ","End":"03:23.905","Text":"The formula for b_n,"},{"Start":"03:23.905 ","End":"03:25.970","Text":"almost the same as the formula for a_n,"},{"Start":"03:25.970 ","End":"03:29.165","Text":"just we have a sine here instead of the cosine."},{"Start":"03:29.165 ","End":"03:33.080","Text":"Once again, we\u0027ll do the substitution."},{"Start":"03:33.080 ","End":"03:36.620","Text":"Well, first of all, f(x) is Pi minus x over 2."},{"Start":"03:36.620 ","End":"03:42.650","Text":"After this substitution, which is the same substitution as we had for a_n,"},{"Start":"03:42.650 ","End":"03:44.915","Text":"so I\u0027ll do it a bit quicker."},{"Start":"03:44.915 ","End":"03:46.735","Text":"This is what we get."},{"Start":"03:46.735 ","End":"03:50.825","Text":"We use again, a trigonometric identity."},{"Start":"03:50.825 ","End":"03:52.460","Text":"Instead cosine of the difference,"},{"Start":"03:52.460 ","End":"03:55.325","Text":"sine of the difference, and this is the formula."},{"Start":"03:55.325 ","End":"03:59.780","Text":"We expand this with the formula and get the following."},{"Start":"03:59.780 ","End":"04:01.955","Text":"Sine of n Pi is 0,"},{"Start":"04:01.955 ","End":"04:06.110","Text":"cosine of n Pi is minus 1^n."},{"Start":"04:06.110 ","End":"04:11.180","Text":"This time we have odd times odd is an even function,"},{"Start":"04:11.180 ","End":"04:13.505","Text":"so it isn\u0027t 0 this time,"},{"Start":"04:13.505 ","End":"04:20.390","Text":"but we can still simplify it by taking the integral from 0 to Pi and then doubling,"},{"Start":"04:20.390 ","End":"04:24.185","Text":"we achieve by just throwing out the 2 here."},{"Start":"04:24.185 ","End":"04:27.050","Text":"What is the integral of t sine nt?"},{"Start":"04:27.050 ","End":"04:32.180","Text":"Let\u0027s do the indefinite integral using integration by parts."},{"Start":"04:32.180 ","End":"04:34.680","Text":"We have t sine nt."},{"Start":"04:34.680 ","End":"04:37.415","Text":"We\u0027ll let this be f and this g\u0027."},{"Start":"04:37.415 ","End":"04:40.729","Text":"g is this, and here it is again,"},{"Start":"04:40.729 ","End":"04:44.570","Text":"and f\u0027 is 1, here\u0027s f again."},{"Start":"04:44.570 ","End":"04:46.190","Text":"This is the formula."},{"Start":"04:46.190 ","End":"04:52.986","Text":"It\u0027s going to be fg minus the integral of f\u0027 g. Now,"},{"Start":"04:52.986 ","End":"04:57.355","Text":"this integral, the integral of cosine is sine."},{"Start":"04:57.355 ","End":"04:59.470","Text":"We also have a minus with a minus,"},{"Start":"04:59.470 ","End":"05:01.150","Text":"so those will cancel out,"},{"Start":"05:01.150 ","End":"05:02.875","Text":"so we get a plus here,"},{"Start":"05:02.875 ","End":"05:05.166","Text":"sine nt but n squared,"},{"Start":"05:05.166 ","End":"05:10.910","Text":"because the integral of cosine nt is sine nt over n,"},{"Start":"05:10.910 ","End":"05:13.240","Text":"plus a constant we should write,"},{"Start":"05:13.240 ","End":"05:15.280","Text":"but we\u0027re going to use it for a definite integral,"},{"Start":"05:15.280 ","End":"05:17.785","Text":"so the constant will cancel."},{"Start":"05:17.785 ","End":"05:19.630","Text":"For this integral,"},{"Start":"05:19.630 ","End":"05:26.260","Text":"just plug in this expression then evaluate it from 0 to Pi."},{"Start":"05:26.260 ","End":"05:29.110","Text":"When t is 0,"},{"Start":"05:29.110 ","End":"05:34.780","Text":"everything is 0 because here t is 0 and here sine nt is 0,"},{"Start":"05:34.780 ","End":"05:36.560","Text":"so the 0 doesn\u0027t matter."},{"Start":"05:36.560 ","End":"05:38.934","Text":"Also when t is Pi,"},{"Start":"05:38.934 ","End":"05:40.840","Text":"sine n Pi is 0,"},{"Start":"05:40.840 ","End":"05:45.740","Text":"so we only need the first term with t equals Pi,"},{"Start":"05:45.740 ","End":"05:52.400","Text":"which comes out to be minus Pi cosine of n Pi."},{"Start":"05:52.400 ","End":"05:55.610","Text":"Now, cosine of n Pi is minus 1^n."},{"Start":"05:55.610 ","End":"05:58.025","Text":"Now, let\u0027s simplify."},{"Start":"05:58.025 ","End":"06:00.395","Text":"Here we have a minus Pi,"},{"Start":"06:00.395 ","End":"06:02.030","Text":"here we have a minus,"},{"Start":"06:02.030 ","End":"06:03.050","Text":"and here we have a Pi,"},{"Start":"06:03.050 ","End":"06:07.970","Text":"so the minus Pi with the minus and the Pi will cancel, also,"},{"Start":"06:07.970 ","End":"06:13.140","Text":"minus 1^n with minus 1^n, they cancel."},{"Start":"06:13.140 ","End":"06:14.300","Text":"Because if you multiply them,"},{"Start":"06:14.300 ","End":"06:17.340","Text":"it\u0027s minus 1 to the 2n, which is 1,"},{"Start":"06:17.340 ","End":"06:22.935","Text":"so all this thing is just 1 over n. This was b_n,"},{"Start":"06:22.935 ","End":"06:25.380","Text":"let\u0027s go back and see."},{"Start":"06:25.380 ","End":"06:29.130","Text":"b_n is all of this,"},{"Start":"06:29.130 ","End":"06:31.580","Text":"so b_n is 1 over n,"},{"Start":"06:31.580 ","End":"06:34.670","Text":"and a_n we had earlier was 0."},{"Start":"06:34.670 ","End":"06:41.810","Text":"Now, we can substitute these in this Fourier representation and get that"},{"Start":"06:41.810 ","End":"06:49.145","Text":"f(x) has a series sine x over n sum n goes from 1 to infinity."},{"Start":"06:49.145 ","End":"06:52.231","Text":"This is because the a_0 is 0, so this cancels, a_n is 0 so this cancels,"},{"Start":"06:52.231 ","End":"06:56.060","Text":"and b_n is 1,"},{"Start":"06:56.060 ","End":"06:58.745","Text":"so we just put n in the denominator here."},{"Start":"06:58.745 ","End":"07:02.000","Text":"This is a series for f(x)."},{"Start":"07:02.000 ","End":"07:04.590","Text":"Now, what about equals."},{"Start":"07:04.590 ","End":"07:08.375","Text":"For x in the interval from 0 to 2 Pi open,"},{"Start":"07:08.375 ","End":"07:11.465","Text":"f(x) is a continuous function."},{"Start":"07:11.465 ","End":"07:14.060","Text":"We can apply Dirichlet\u0027s theorem."},{"Start":"07:14.060 ","End":"07:17.390","Text":"That proves the first part of the question."},{"Start":"07:17.390 ","End":"07:22.375","Text":"Then there was another part which asked what happens at the end points."},{"Start":"07:22.375 ","End":"07:25.120","Text":"If x is 0 or 2 Pi,"},{"Start":"07:25.120 ","End":"07:32.675","Text":"then the series converges to the average of the 2 endpoints,"},{"Start":"07:32.675 ","End":"07:37.324","Text":"0 from the right and 2 Pi from the left."},{"Start":"07:37.324 ","End":"07:41.090","Text":"Pi-x/2 once with x=0,"},{"Start":"07:41.090 ","End":"07:43.070","Text":"once with x=2 Pi,"},{"Start":"07:43.070 ","End":"07:44.990","Text":"and the average of these 2."},{"Start":"07:44.990 ","End":"07:47.155","Text":"This is Pi/2."},{"Start":"07:47.155 ","End":"07:50.535","Text":"This is minus Pi over 2,"},{"Start":"07:50.535 ","End":"07:53.972","Text":"that\u0027s the value here is Pi/2,"},{"Start":"07:53.972 ","End":"07:55.880","Text":"the value here is minus Pi over 2,"},{"Start":"07:55.880 ","End":"07:58.340","Text":"if you add this and this and divide it by 2,"},{"Start":"07:58.340 ","End":"08:03.050","Text":"you get 0 is a revised diagram,"},{"Start":"08:03.050 ","End":"08:04.670","Text":"included a bit more."},{"Start":"08:04.670 ","End":"08:08.510","Text":"I\u0027ve also put in these points,"},{"Start":"08:08.510 ","End":"08:12.730","Text":"which is what the series converges to."},{"Start":"08:12.730 ","End":"08:16.360","Text":"I also extended this periodically."},{"Start":"08:16.360 ","End":"08:24.260","Text":"What happens at 0 is that here it\u0027s Pi and from the left it\u0027s minus Pi,"},{"Start":"08:24.260 ","End":"08:27.325","Text":"and actually, turns out to be the average."},{"Start":"08:27.325 ","End":"08:30.755","Text":"Similarly, here at 2 Pi,"},{"Start":"08:30.755 ","End":"08:33.875","Text":"it\u0027s 0, it\u0027s the average."},{"Start":"08:33.875 ","End":"08:35.870","Text":"This actually makes sense."},{"Start":"08:35.870 ","End":"08:37.685","Text":"Just to check,"},{"Start":"08:37.685 ","End":"08:40.475","Text":"if we put x equals 0,"},{"Start":"08:40.475 ","End":"08:43.940","Text":"then we have sine of 0."},{"Start":"08:43.940 ","End":"08:46.190","Text":"The whole series becomes 0,"},{"Start":"08:46.190 ","End":"08:47.840","Text":"so we have the sum of 0\u0027s,"},{"Start":"08:47.840 ","End":"08:49.955","Text":"which is 0, and that\u0027s fine."},{"Start":"08:49.955 ","End":"08:52.460","Text":"If we put x equals 2 Pi,"},{"Start":"08:52.460 ","End":"08:55.745","Text":"then we have sine of 2 and Pi is also 0."},{"Start":"08:55.745 ","End":"09:00.680","Text":"Again, we have sum of all 0\u0027s that will give us 0."},{"Start":"09:00.680 ","End":"09:04.840","Text":"It makes sense that the series converges here to 0 and here to 0."},{"Start":"09:04.840 ","End":"09:07.910","Text":"In general, whenever there\u0027s this jump discontinuity,"},{"Start":"09:07.910 ","End":"09:11.290","Text":"the Fourier series takes the average value."},{"Start":"09:11.290 ","End":"09:13.930","Text":"That\u0027s it for Part A."}],"ID":28774},{"Watched":false,"Name":"Exercise 5 - Part b","Duration":"2m 42s","ChapterTopicVideoID":27611,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.385","Text":"Now we come to Part B."},{"Start":"00:02.385 ","End":"00:06.990","Text":"We just proved in Part A that the sum of this series is this,"},{"Start":"00:06.990 ","End":"00:10.050","Text":"and now we have a variant of this."},{"Start":"00:10.050 ","End":"00:13.275","Text":"Instead of the n here we have n^3."},{"Start":"00:13.275 ","End":"00:17.361","Text":"We have to compute this sum explicitly,"},{"Start":"00:17.361 ","End":"00:18.600","Text":"not as a series,"},{"Start":"00:18.600 ","End":"00:22.500","Text":"and we\u0027re given a hint to use term-by-term integration."},{"Start":"00:22.500 ","End":"00:26.145","Text":"Actually, we\u0027re going to use term-by-term integration twice."},{"Start":"00:26.145 ","End":"00:27.581","Text":"Well, you\u0027ll see."},{"Start":"00:27.581 ","End":"00:31.605","Text":"Note that if we take the indefinite integral of sine nx,"},{"Start":"00:31.605 ","End":"00:35.100","Text":"we get minus cosine nx over n^2."},{"Start":"00:35.100 ","End":"00:37.950","Text":"If we take the indefinite integral of this,"},{"Start":"00:37.950 ","End":"00:44.675","Text":"we get minus sine x over n^3 which is just like this with an extra minus."},{"Start":"00:44.675 ","End":"00:49.445","Text":"That\u0027s what gives the plan of doing term-by-term integration twice."},{"Start":"00:49.445 ","End":"00:51.410","Text":"We start with this series,"},{"Start":"00:51.410 ","End":"00:53.270","Text":"and then integrate it,"},{"Start":"00:53.270 ","End":"00:55.625","Text":"and we do an indefinite integral."},{"Start":"00:55.625 ","End":"00:57.565","Text":"That\u0027s why we need a constant."},{"Start":"00:57.565 ","End":"01:00.650","Text":"The usual trick is not to let it be C,"},{"Start":"01:00.650 ","End":"01:04.310","Text":"but a_0/2 just like in the Fourier series."},{"Start":"01:04.310 ","End":"01:08.720","Text":"Then we have a formula for a_0 which is this."},{"Start":"01:08.720 ","End":"01:12.650","Text":"You may be used to minus Pi-Pi for the integral,"},{"Start":"01:12.650 ","End":"01:15.680","Text":"but here we\u0027re not on the interval minus Pi, Pi."},{"Start":"01:15.680 ","End":"01:18.355","Text":"We\u0027re on the interval 0(2Pi)."},{"Start":"01:18.355 ","End":"01:21.485","Text":"Now this is a polynomial straightforward integral."},{"Start":"01:21.485 ","End":"01:25.505","Text":"This is what it is, just plug in 0 and 2Pi,"},{"Start":"01:25.505 ","End":"01:27.530","Text":"0 at 0 at 2 Pi."},{"Start":"01:27.530 ","End":"01:29.450","Text":"Well, just a simple substitution."},{"Start":"01:29.450 ","End":"01:34.740","Text":"This is what we get and that boils down to 1/3 Pi^2."},{"Start":"01:34.740 ","End":"01:40.230","Text":"Now if we put this a_0 into here and we get Pi^2 over 6,"},{"Start":"01:40.230 ","End":"01:42.950","Text":"so we get this."},{"Start":"01:42.950 ","End":"01:47.640","Text":"Now, bring this constant over to the other side multiply by minus 1,"},{"Start":"01:47.640 ","End":"01:54.305","Text":"then we get a formula for this sum of cosine of x over n^2 is the following polynomial."},{"Start":"01:54.305 ","End":"01:57.755","Text":"We\u0027re going to do another term-by-term integration."},{"Start":"01:57.755 ","End":"02:04.370","Text":"This time we\u0027ll get sine nx over n^3 because that\u0027s what the integral of cosine nx is;"},{"Start":"02:04.370 ","End":"02:05.660","Text":"sine nx over n,"},{"Start":"02:05.660 ","End":"02:10.100","Text":"and we need the integral of this element to do the indefinite integral."},{"Start":"02:10.100 ","End":"02:15.005","Text":"This is again a straightforward integral and this is what it is."},{"Start":"02:15.005 ","End":"02:17.120","Text":"You have to add a constant of integration,"},{"Start":"02:17.120 ","End":"02:20.420","Text":"but that constant comes out to be 0 because if we plug"},{"Start":"02:20.420 ","End":"02:23.890","Text":"in x equals 0 all these signs are 0. So the sum is 0."},{"Start":"02:23.890 ","End":"02:25.370","Text":"Also this is 0,"},{"Start":"02:25.370 ","End":"02:28.370","Text":"this is 0, this is 0, so C is 0."},{"Start":"02:28.370 ","End":"02:35.525","Text":"This is the series for sine x over n^3,"},{"Start":"02:35.525 ","End":"02:43.110","Text":"and that means that this is the g(x) that we were looking for and that concludes Part B."}],"ID":28775},{"Watched":false,"Name":"Exercise 6","Duration":"4m 7s","ChapterTopicVideoID":27612,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this exercise, we have a function on minus Pi,"},{"Start":"00:03.780 ","End":"00:09.405","Text":"Pi, which is k minus 1 times continuously differentiable."},{"Start":"00:09.405 ","End":"00:15.705","Text":"K is some positive integer and k times piecewise continuously differentiable."},{"Start":"00:15.705 ","End":"00:20.970","Text":"So first k minus 1 derivatives are continuous."},{"Start":"00:20.970 ","End":"00:26.235","Text":"The last time we only get something that\u0027s piecewise continuously differentiable."},{"Start":"00:26.235 ","End":"00:31.290","Text":"Suppose that the derivatives up to all the k minus"},{"Start":"00:31.290 ","End":"00:37.109","Text":"1 have the property that at minus Pi and at Pi they\u0027re equal,"},{"Start":"00:37.109 ","End":"00:42.120","Text":"which means that they could be extended periodically to the whole real line."},{"Start":"00:42.120 ","End":"00:49.780","Text":"Now we suppose that the complex Fourier expansion of f is sum c_n e^inx,"},{"Start":"00:49.780 ","End":"00:51.260","Text":"all this is what\u0027s given."},{"Start":"00:51.260 ","End":"00:58.760","Text":"We have to prove that the limit of n^k c_n is 0. Well, let\u0027s solve it."},{"Start":"00:58.760 ","End":"01:03.785","Text":"The main step is to prove by induction the following claim,"},{"Start":"01:03.785 ","End":"01:08.645","Text":"that for each j from 0 up to k,"},{"Start":"01:08.645 ","End":"01:15.365","Text":"the jth derivative of f is given by the following formula,"},{"Start":"01:15.365 ","End":"01:22.280","Text":"which basically means that we differentiate the series term by term j times."},{"Start":"01:22.280 ","End":"01:27.125","Text":"Let\u0027s start the induction with j=0, the base case."},{"Start":"01:27.125 ","End":"01:28.940","Text":"We have to prove the following."},{"Start":"01:28.940 ","End":"01:33.230","Text":"Just replace j by 0 and the 0th derivative is just the function"},{"Start":"01:33.230 ","End":"01:37.810","Text":"itself so we have here that f(x) is equal to and this is 1."},{"Start":"01:37.810 ","End":"01:39.210","Text":"We have to prove this,"},{"Start":"01:39.210 ","End":"01:41.545","Text":"but this is what\u0027s given."},{"Start":"01:41.545 ","End":"01:43.895","Text":"Yeah, here it is."},{"Start":"01:43.895 ","End":"01:45.995","Text":"Now the induction step."},{"Start":"01:45.995 ","End":"01:50.900","Text":"Suppose that j is less than k. We\u0027re going to show that if it\u0027s true for j,"},{"Start":"01:50.900 ","End":"01:52.930","Text":"it\u0027s true for j plus 1."},{"Start":"01:52.930 ","End":"01:55.140","Text":"But we don\u0027t want to pass k,"},{"Start":"01:55.140 ","End":"01:58.710","Text":"we\u0027re only guaranteed up to k so we suppose that j is less than"},{"Start":"01:58.710 ","End":"02:02.705","Text":"k. We have that this is true, the induction hypothesis."},{"Start":"02:02.705 ","End":"02:06.545","Text":"We have to prove that if you replace j by j plus 1 here,"},{"Start":"02:06.545 ","End":"02:08.375","Text":"it\u0027s also going to be true."},{"Start":"02:08.375 ","End":"02:17.465","Text":"Let big F be the jth derivative of f. We have that f of minus Pi equals f of Pi,"},{"Start":"02:17.465 ","End":"02:20.975","Text":"and that f is piecewise continuous."},{"Start":"02:20.975 ","End":"02:26.415","Text":"This is because we\u0027re told that it\u0027s k minus 1 times continuously differentiable,"},{"Start":"02:26.415 ","End":"02:28.130","Text":"and we know that f\u0027,"},{"Start":"02:28.130 ","End":"02:30.500","Text":"which is derivative j plus 1,"},{"Start":"02:30.500 ","End":"02:34.730","Text":"we know that this is piecewise continuously differentiable."},{"Start":"02:34.730 ","End":"02:38.300","Text":"We don\u0027t know about the last one if it\u0027s equal to k on not."},{"Start":"02:38.300 ","End":"02:42.860","Text":"All we can guarantee is piecewise continuously differentiable. But that\u0027s enough."},{"Start":"02:42.860 ","End":"02:48.660","Text":"We\u0027re going to apply the theorem on term-by-term differentiation."},{"Start":"02:48.660 ","End":"02:52.955","Text":"This theorem here, you can pause and study this, you should know this."},{"Start":"02:52.955 ","End":"02:57.680","Text":"What we get is that this is differentiable and"},{"Start":"02:57.680 ","End":"03:02.555","Text":"the derivative is the term-by-term derivative of the series,"},{"Start":"03:02.555 ","End":"03:06.770","Text":"which gives us an extra power in the in to the power of."},{"Start":"03:06.770 ","End":"03:09.260","Text":"To the j we have j plus 1."},{"Start":"03:09.260 ","End":"03:13.805","Text":"Of course, the derivative of big F will be the j plus 1,"},{"Start":"03:13.805 ","End":"03:17.210","Text":"or j plus first derivative of little f,"},{"Start":"03:17.210 ","End":"03:21.975","Text":"because f is j times derivative, so 1 more."},{"Start":"03:21.975 ","End":"03:24.970","Text":"That completes the proof by induction."},{"Start":"03:24.970 ","End":"03:28.420","Text":"We know that this is true for j equals 0,1,2"},{"Start":"03:28.420 ","End":"03:31.610","Text":"up to k. Now let\u0027s just take the case for k,"},{"Start":"03:31.610 ","End":"03:33.385","Text":"that\u0027s all we really need."},{"Start":"03:33.385 ","End":"03:42.215","Text":"We get that the kth derivative of f is equal to this sum in^k, c_n e^inx."},{"Start":"03:42.215 ","End":"03:46.395","Text":"Next, we\u0027re going to apply the Riemann Lebesgue Lemma to this"},{"Start":"03:46.395 ","End":"03:50.900","Text":"and we get that the limit of the kth coefficient,"},{"Start":"03:50.900 ","End":"03:54.745","Text":"the limit as n goes to infinity of this is 0."},{"Start":"03:54.745 ","End":"03:58.280","Text":"Because the k is the constant,"},{"Start":"03:58.280 ","End":"04:03.140","Text":"we can just take the constant out in front so what we get is i^k times 0,"},{"Start":"04:03.140 ","End":"04:04.685","Text":"which is still 0."},{"Start":"04:04.685 ","End":"04:06.200","Text":"This is what we had to prove,"},{"Start":"04:06.200 ","End":"04:08.370","Text":"and so we are done."}],"ID":28776},{"Watched":false,"Name":"Exercise 7 - Part a","Duration":"3m 51s","ChapterTopicVideoID":27613,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.665","Text":"In this exercise, we have 2 parts which are related."},{"Start":"00:04.665 ","End":"00:06.735","Text":"In part a, we have f,"},{"Start":"00:06.735 ","End":"00:11.595","Text":"which is continuously differentiable on minus Pi Pi,"},{"Start":"00:11.595 ","End":"00:15.735","Text":"and it\u0027s equal at the endpoints,"},{"Start":"00:15.735 ","End":"00:18.240","Text":"and the integral is 0."},{"Start":"00:18.240 ","End":"00:20.955","Text":"We have to prove this inequality."},{"Start":"00:20.955 ","End":"00:25.110","Text":"The integral of the norm of f^2 is less"},{"Start":"00:25.110 ","End":"00:29.820","Text":"than or equal to the integral of the norm of f\u0027^2."},{"Start":"00:29.820 ","End":"00:31.920","Text":"Part B is very similar."},{"Start":"00:31.920 ","End":"00:33.705","Text":"Instead of minus Pi Pi,"},{"Start":"00:33.705 ","End":"00:35.625","Text":"we have 0 Pi,"},{"Start":"00:35.625 ","End":"00:38.010","Text":"and instead of these 2 conditions,"},{"Start":"00:38.010 ","End":"00:40.440","Text":"equality at the endpoint and the integral is 0,"},{"Start":"00:40.440 ","End":"00:42.135","Text":"we have a different condition."},{"Start":"00:42.135 ","End":"00:45.375","Text":"Just that f(0) is f(Pi) is 0."},{"Start":"00:45.375 ","End":"00:51.280","Text":"We have to prove the analogous inequality just instead of minus Pi Pi from 0 to Pi."},{"Start":"00:51.280 ","End":"00:53.385","Text":"Anyway, let\u0027s start with a."},{"Start":"00:53.385 ","End":"00:59.120","Text":"Now, f satisfies the condition of the theorem on term by term differentiation,"},{"Start":"00:59.120 ","End":"01:00.575","Text":"and I\u0027ll remind you."},{"Start":"01:00.575 ","End":"01:05.360","Text":"Here it is. The conditions are that f is continuous and equal at"},{"Start":"01:05.360 ","End":"01:11.255","Text":"the endpoints and it\u0027s piecewise continuously differentiable."},{"Start":"01:11.255 ","End":"01:13.460","Text":"Back to our exercise."},{"Start":"01:13.460 ","End":"01:14.930","Text":"F satisfies the condition,"},{"Start":"01:14.930 ","End":"01:18.260","Text":"so we can use the conclusions of the theorem."},{"Start":"01:18.260 ","End":"01:22.610","Text":"That is, if f has a Fourier series like this,"},{"Start":"01:22.610 ","End":"01:25.610","Text":"then f\u0027 will have a Fourier series which is"},{"Start":"01:25.610 ","End":"01:29.120","Text":"a term by term differentiation of this series."},{"Start":"01:29.120 ","End":"01:32.180","Text":"Then we take each coefficient and multiply it by in."},{"Start":"01:32.180 ","End":"01:35.825","Text":"Now let\u0027s apply Parseval\u0027s identity."},{"Start":"01:35.825 ","End":"01:45.730","Text":"The integral of absolute value of f^2=2Pi times the sum of this series,"},{"Start":"01:45.730 ","End":"01:48.920","Text":"where we take the absolute value of the coefficient squared."},{"Start":"01:48.920 ","End":"01:53.040","Text":"It\u0027s written usually with 2Pi on the other side,"},{"Start":"01:53.040 ","End":"01:56.325","Text":"1 over 2Pi, but I just brought the 2Pi over here."},{"Start":"01:56.325 ","End":"01:59.760","Text":"That\u0027s for f. Similarly for f\u0027,"},{"Start":"01:59.760 ","End":"02:05.215","Text":"2Pi times the sum of the absolute value of the coefficient squared,"},{"Start":"02:05.215 ","End":"02:09.635","Text":"so it\u0027s just like the c_n except we have an in there also."},{"Start":"02:09.635 ","End":"02:14.330","Text":"What we have to prove is this just copied it here."},{"Start":"02:14.330 ","End":"02:16.460","Text":"Now to prove this,"},{"Start":"02:16.460 ","End":"02:19.910","Text":"it\u0027s enough to show that the absolute value of"},{"Start":"02:19.910 ","End":"02:24.350","Text":"c_n is less than or equal to the absolute value of inc_n."},{"Start":"02:24.350 ","End":"02:25.820","Text":"If this is true,"},{"Start":"02:25.820 ","End":"02:28.115","Text":"then when you square it, it\u0027s true."},{"Start":"02:28.115 ","End":"02:31.985","Text":"Also note that we can throw out the i here,"},{"Start":"02:31.985 ","End":"02:34.985","Text":"so we have to show basically that for each n,"},{"Start":"02:34.985 ","End":"02:40.020","Text":"absolute value of c_n is less than or equal to absolute value of nc_n."},{"Start":"02:40.020 ","End":"02:43.595","Text":"Take 2 cases, n=0, n not equal to 0."},{"Start":"02:43.595 ","End":"02:45.575","Text":"If n is not equal to 0,"},{"Start":"02:45.575 ","End":"02:49.445","Text":"it\u0027s an integer, so its absolute value is at least 1."},{"Start":"02:49.445 ","End":"02:53.870","Text":"Absolute value of c_n is less than or equal to nc_n because this is"},{"Start":"02:53.870 ","End":"02:56.120","Text":"the absolute value of n times absolute value of"},{"Start":"02:56.120 ","End":"02:58.625","Text":"c_n and absolute value of c_n is non negative."},{"Start":"02:58.625 ","End":"03:00.380","Text":"Anyway, we get this,"},{"Start":"03:00.380 ","End":"03:01.850","Text":"that\u0027s the 1 case."},{"Start":"03:01.850 ","End":"03:05.350","Text":"The other case is where n=0,"},{"Start":"03:05.350 ","End":"03:08.000","Text":"so we\u0027re trying to show that absolute value of c_0"},{"Start":"03:08.000 ","End":"03:11.800","Text":"is less than or equal to absolute value of 0c_0."},{"Start":"03:11.800 ","End":"03:15.330","Text":"Which essentially means that c_0 is 0."},{"Start":"03:15.330 ","End":"03:17.750","Text":"The absolute value has to be less than or equal to 0,"},{"Start":"03:17.750 ","End":"03:19.580","Text":"so it has to be equal to 0."},{"Start":"03:19.580 ","End":"03:21.325","Text":"We\u0027ll show this."},{"Start":"03:21.325 ","End":"03:24.880","Text":"c_0 using the formula is this integral,"},{"Start":"03:24.880 ","End":"03:27.103","Text":"same formula as for the other ends,"},{"Start":"03:27.103 ","End":"03:30.530","Text":"each of the minus i and x here."},{"Start":"03:30.530 ","End":"03:36.715","Text":"This is equal to 1, so it\u0027s just integral of f(x)dx and we\u0027re given that this is 0."},{"Start":"03:36.715 ","End":"03:40.145","Text":"The 1 over 2Pi doesn\u0027t affect it to 0."},{"Start":"03:40.145 ","End":"03:42.930","Text":"We\u0027ve shown it for both cases."},{"Start":"03:43.090 ","End":"03:46.055","Text":"Basically, we\u0027ve shown everything we need,"},{"Start":"03:46.055 ","End":"03:48.965","Text":"the sufficient condition for our inequality to hold."},{"Start":"03:48.965 ","End":"03:52.410","Text":"That concludes Part A."}],"ID":28777},{"Watched":false,"Name":"Exercise 7 - Part b","Duration":"4m 24s","ChapterTopicVideoID":27614,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.220","Text":"Now we come to Part b of the exercise."},{"Start":"00:05.220 ","End":"00:08.025","Text":"We\u0027ve just done part a and in fact,"},{"Start":"00:08.025 ","End":"00:10.935","Text":"we\u0027ll solve b by reducing it to a."},{"Start":"00:10.935 ","End":"00:16.425","Text":"Just to remind you, we have different conditions on f. This time it\u0027s defined only on 0,"},{"Start":"00:16.425 ","End":"00:20.940","Text":"Pi and it has the property that at the 2 endpoints,"},{"Start":"00:20.940 ","End":"00:23.100","Text":"0 and Pi, not only they\u0027re equal,"},{"Start":"00:23.100 ","End":"00:24.660","Text":"but it\u0027s equal to 0."},{"Start":"00:24.660 ","End":"00:27.540","Text":"We have to prove essentially the same inequality"},{"Start":"00:27.540 ","End":"00:31.110","Text":"except that this time on the interval from 0 to Pi."},{"Start":"00:31.110 ","End":"00:32.910","Text":"The plan will be,"},{"Start":"00:32.910 ","End":"00:36.740","Text":"that we will extend our little f that\u0027s defined on 0,"},{"Start":"00:36.740 ","End":"00:40.905","Text":"Pi to another function big F on minus Pi, Pi."},{"Start":"00:40.905 ","End":"00:47.810","Text":"We\u0027ll show that big F satisfies the conditions that little f satisfied in part a,"},{"Start":"00:47.810 ","End":"00:49.325","Text":"it will take the role of it."},{"Start":"00:49.325 ","End":"00:51.770","Text":"Then assuming that we\u0027ve done this,"},{"Start":"00:51.770 ","End":"00:53.525","Text":"let\u0027s see how we would proceed."},{"Start":"00:53.525 ","End":"00:59.195","Text":"We have the integral of absolute value of f^2 is the integral of"},{"Start":"00:59.195 ","End":"01:06.410","Text":"big F^2 absolute value because little f is equal to big F when you extend something,"},{"Start":"01:06.410 ","End":"01:08.690","Text":"it stays the same on the original part,"},{"Start":"01:08.690 ","End":"01:10.820","Text":"regardless how you extend it."},{"Start":"01:10.820 ","End":"01:12.589","Text":"F is an odd function."},{"Start":"01:12.589 ","End":"01:15.950","Text":"When it takes the absolute value, becomes even."},{"Start":"01:15.950 ","End":"01:18.355","Text":"If you square it, it\u0027s still even."},{"Start":"01:18.355 ","End":"01:24.200","Text":"This is equal to half the integral on the whole interval minus Pi, Pi."},{"Start":"01:24.200 ","End":"01:27.140","Text":"Usually, we do it the other way round that this is twice this."},{"Start":"01:27.140 ","End":"01:28.730","Text":"But okay works the other way."},{"Start":"01:28.730 ","End":"01:31.700","Text":"Now using the result from part a,"},{"Start":"01:31.700 ","End":"01:35.870","Text":"we can replace big F by big F\u0027,"},{"Start":"01:35.870 ","End":"01:39.650","Text":"and then we just do the same thing as here, but backwards."},{"Start":"01:39.650 ","End":"01:41.720","Text":"This is also even."},{"Start":"01:41.720 ","End":"01:44.615","Text":"The derivative of an odd function is an even function."},{"Start":"01:44.615 ","End":"01:46.310","Text":"When we take the absolute value and square it,"},{"Start":"01:46.310 ","End":"01:50.035","Text":"it\u0027s still even so a half of this is equal to this."},{"Start":"01:50.035 ","End":"01:53.000","Text":"Then on the interval from 0 to Pi,"},{"Start":"01:53.000 ","End":"01:58.550","Text":"big F is equal to little f and so the derivatives are also equal so we get this."},{"Start":"01:58.550 ","End":"02:00.230","Text":"If you look at the beginning and the end,"},{"Start":"02:00.230 ","End":"02:02.210","Text":"then that\u0027s exactly what we want."},{"Start":"02:02.210 ","End":"02:05.900","Text":"Everything\u0027s equals except for here we have a less than or equal to."},{"Start":"02:05.900 ","End":"02:10.025","Text":"What remains is to show that big F satisfies the conditions of a,"},{"Start":"02:10.025 ","End":"02:13.025","Text":"meaning the conditions that little f satisfied there."},{"Start":"02:13.025 ","End":"02:17.135","Text":"Reminder of what the odd extension is."},{"Start":"02:17.135 ","End":"02:24.550","Text":"We define it as the original function between 0 and Pi and minus f(minus x) here."},{"Start":"02:24.550 ","End":"02:27.500","Text":"We\u0027ll see why we had to take odd and not,"},{"Start":"02:27.500 ","End":"02:29.960","Text":"for example, the even extension."},{"Start":"02:29.960 ","End":"02:32.980","Text":"Well, here\u0027s the picture of what the odd extension is."},{"Start":"02:32.980 ","End":"02:34.480","Text":"We had it from 0 to Pi."},{"Start":"02:34.480 ","End":"02:36.294","Text":"Let\u0027s say when we extended,"},{"Start":"02:36.294 ","End":"02:39.055","Text":"oddly than it\u0027s this part."},{"Start":"02:39.055 ","End":"02:41.050","Text":"The even extension would be this,"},{"Start":"02:41.050 ","End":"02:42.700","Text":"which doesn\u0027t work as we\u0027ll see."},{"Start":"02:42.700 ","End":"02:48.280","Text":"Now, this is actually not a good definition because 0 appears both places."},{"Start":"02:48.280 ","End":"02:51.220","Text":"If we take the 0 and plug it in here,"},{"Start":"02:51.220 ","End":"02:57.475","Text":"we get minus f(minus 0) well minus 0 is the same as 0 so it\u0027s minus f(0)."},{"Start":"02:57.475 ","End":"03:00.990","Text":"We know that f(0) is 0."},{"Start":"03:00.990 ","End":"03:02.670","Text":"That was given, here it is."},{"Start":"03:02.670 ","End":"03:05.550","Text":"f(0) is 0 and that\u0027s 0 and that\u0027s f(0)."},{"Start":"03:05.550 ","End":"03:09.730","Text":"We get this equals this so it\u0027s well-defined."},{"Start":"03:09.730 ","End":"03:14.450","Text":"But it also means that the function is continuous."},{"Start":"03:14.450 ","End":"03:17.435","Text":"It\u0027s continuous here, continuous here,"},{"Start":"03:17.435 ","End":"03:25.230","Text":"and matches at this point here so it\u0027s continuous on the whole minus Pi-Pi."},{"Start":"03:25.230 ","End":"03:27.890","Text":"Also note that the integral is 0 because when we"},{"Start":"03:27.890 ","End":"03:30.800","Text":"have an odd function on a symmetric interval,"},{"Start":"03:30.800 ","End":"03:32.545","Text":"the integral is 0."},{"Start":"03:32.545 ","End":"03:33.859","Text":"One of the conditions,"},{"Start":"03:33.859 ","End":"03:35.930","Text":"the one that\u0027s remaining is to show that f is"},{"Start":"03:35.930 ","End":"03:39.110","Text":"continuously differentiable on the interval."},{"Start":"03:39.110 ","End":"03:41.690","Text":"It\u0027s continuously differentiable here and continuously"},{"Start":"03:41.690 ","End":"03:44.090","Text":"differentiable here and even continuous here."},{"Start":"03:44.090 ","End":"03:45.920","Text":"We just have to show that"},{"Start":"03:45.920 ","End":"03:50.630","Text":"the derivative limit from the right and from the left are equal."},{"Start":"03:50.630 ","End":"03:52.070","Text":"Well, I\u0027ll leave you to check this."},{"Start":"03:52.070 ","End":"03:55.805","Text":"It\u0027s sort of clear from the picture that whenever you have an odd function and"},{"Start":"03:55.805 ","End":"03:59.675","Text":"you flip it 180 degrees around this point here,"},{"Start":"03:59.675 ","End":"04:01.285","Text":"it goes onto itself."},{"Start":"04:01.285 ","End":"04:05.900","Text":"It\u0027s obvious that the slope here becomes the slope here."},{"Start":"04:05.900 ","End":"04:07.735","Text":"Anyway, here it is written down."},{"Start":"04:07.735 ","End":"04:12.920","Text":"We have that f\u0027 is differentiable at 0 and ends on the whole ( minus Pi, Pi)."},{"Start":"04:12.920 ","End":"04:16.025","Text":"Now, big F has met all the conditions that we need,"},{"Start":"04:16.025 ","End":"04:19.520","Text":"and that\u0027s all that we had remaining to prove because from here,"},{"Start":"04:19.520 ","End":"04:25.160","Text":"we already did that part at the beginning. We are done."}],"ID":28778},{"Watched":false,"Name":"Exercise 8 - Part a","Duration":"3m 4s","ChapterTopicVideoID":27615,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise, we have the function f(x),"},{"Start":"00:03.480 ","End":"00:11.175","Text":"which is the sum of a series of these functions and this is on all of the real line."},{"Start":"00:11.175 ","End":"00:14.520","Text":"Part a, we have to prove that f is continuous."},{"Start":"00:14.520 ","End":"00:21.060","Text":"In part b, we have to prove that f is not continuously differentiable."},{"Start":"00:21.060 ","End":"00:24.780","Text":"There\u0027s something not quite right or not precise."},{"Start":"00:24.780 ","End":"00:25.980","Text":"When I say equals,"},{"Start":"00:25.980 ","End":"00:29.940","Text":"this assumes that this converges to finite sum,"},{"Start":"00:29.940 ","End":"00:32.100","Text":"I mean they\u0027re all positive terms,"},{"Start":"00:32.100 ","End":"00:33.720","Text":"but it could be infinity."},{"Start":"00:33.720 ","End":"00:39.000","Text":"We\u0027ll see during the solution of part a that this really does converge."},{"Start":"00:39.000 ","End":"00:43.050","Text":"In fact, it converges uniformly so that f really exists,"},{"Start":"00:43.050 ","End":"00:44.855","Text":"it\u0027s not infinity anywhere."},{"Start":"00:44.855 ","End":"00:48.635","Text":"Let\u0027s start with part a, that each term,"},{"Start":"00:48.635 ","End":"00:55.445","Text":"each general term be called f_n so that f(x) is the sum of f_n(x)."},{"Start":"00:55.445 ","End":"00:59.630","Text":"Now each of this f_n is clearly continuous."},{"Start":"00:59.630 ","End":"01:03.845","Text":"If we show that this series converges uniformly,"},{"Start":"01:03.845 ","End":"01:07.490","Text":"then we can conclude that the sum is also continuous."},{"Start":"01:07.490 ","End":"01:11.600","Text":"Reminder, if we have a sequence of continuous functions on"},{"Start":"01:11.600 ","End":"01:17.945","Text":"some domain and the sum of the series converges uniformly to the sum f,"},{"Start":"01:17.945 ","End":"01:22.160","Text":"then f is also continuous on the domain."},{"Start":"01:22.160 ","End":"01:24.800","Text":"We\u0027ll use the Weierstrass M-test,"},{"Start":"01:24.800 ","End":"01:27.665","Text":"which is the main tool we use in this situation."},{"Start":"01:27.665 ","End":"01:30.620","Text":"I\u0027ll remind you what the Weierstrass M-test is."},{"Start":"01:30.620 ","End":"01:33.320","Text":"I copy-pasted this from the Wikipedia."},{"Start":"01:33.320 ","End":"01:39.065","Text":"Basically, we have to show that the terms f_n(x) are bounded by"},{"Start":"01:39.065 ","End":"01:45.170","Text":"constants m_n and that the sum of these constants m_n is less than infinity,"},{"Start":"01:45.170 ","End":"01:48.080","Text":"that it\u0027s a convergence series of numbers."},{"Start":"01:48.080 ","End":"01:55.030","Text":"The absolute value of f_n is the absolute value of 1 over 1 plus x^2 e^i, n^2 x."},{"Start":"01:55.030 ","End":"01:56.940","Text":"Now, each of the in^2,"},{"Start":"01:56.940 ","End":"02:01.190","Text":"x is e^i times something real so the absolute value is 1."},{"Start":"02:01.190 ","End":"02:03.350","Text":"This is always on the unit circle."},{"Start":"02:03.350 ","End":"02:06.845","Text":"This is just the absolute value of 1 over 1 plus n^2 and"},{"Start":"02:06.845 ","End":"02:10.399","Text":"we can drop the absolute value because this is positive,"},{"Start":"02:10.399 ","End":"02:13.350","Text":"and we\u0027ll define this to be m_n,"},{"Start":"02:13.350 ","End":"02:17.600","Text":"should really use this symbol because it\u0027s a definition not just an inequality."},{"Start":"02:17.600 ","End":"02:22.400","Text":"The sum of the m_n is less than or equal to the sum of 1 over"},{"Start":"02:22.400 ","End":"02:27.420","Text":"n^2 because 1 over 1 plus n^2 is less than 1 over n^2."},{"Start":"02:27.420 ","End":"02:29.630","Text":"You decrease the denominator,"},{"Start":"02:29.630 ","End":"02:31.700","Text":"you increase the fraction."},{"Start":"02:31.700 ","End":"02:34.700","Text":"This sum is a well-known sum."},{"Start":"02:34.700 ","End":"02:37.670","Text":"It\u0027s a p-series with p=2,"},{"Start":"02:37.670 ","End":"02:41.360","Text":"or we just know that sum is Pi^2 over 6."},{"Start":"02:41.360 ","End":"02:45.540","Text":"Anyway, it\u0027s finite so we\u0027re okay there."},{"Start":"02:45.540 ","End":"02:51.105","Text":"It means that the M-test can be applied and the sum of f_n,"},{"Start":"02:51.105 ","End":"02:54.345","Text":"which is f, is a uniform sum."},{"Start":"02:54.345 ","End":"02:56.870","Text":"Because this convergence is uniform,"},{"Start":"02:56.870 ","End":"03:01.490","Text":"then the sum f is also continuous and that\u0027s what we had to show,"},{"Start":"03:01.490 ","End":"03:04.500","Text":"and that concludes part a."}],"ID":28779},{"Watched":false,"Name":"Exercise 8 - Part b","Duration":"2m 35s","ChapterTopicVideoID":27616,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"Now, we come to part b."},{"Start":"00:02.940 ","End":"00:07.200","Text":"We just did part a where we showed that f is continuous."},{"Start":"00:07.200 ","End":"00:11.310","Text":"Now we\u0027re going to show that f is not continuously differentiable."},{"Start":"00:11.310 ","End":"00:16.440","Text":"Let\u0027s restrict f meanwhile to the interval minus Pi to Pi."},{"Start":"00:16.440 ","End":"00:22.080","Text":"Note that the Fourier series of f is just this series."},{"Start":"00:22.080 ","End":"00:27.435","Text":"If f is equal to the sum of a Fourier series,"},{"Start":"00:27.435 ","End":"00:34.705","Text":"then this Fourier series is the Fourier series for the function f, its representation."},{"Start":"00:34.705 ","End":"00:36.330","Text":"It doesn\u0027t work the other way around."},{"Start":"00:36.330 ","End":"00:38.640","Text":"We can\u0027t conclude from this the equality,"},{"Start":"00:38.640 ","End":"00:39.960","Text":"but if it\u0027s equal,"},{"Start":"00:39.960 ","End":"00:42.615","Text":"that\u0027s also represented by."},{"Start":"00:42.615 ","End":"00:47.000","Text":"Now, the plan is to apply term by term differentiation to this."},{"Start":"00:47.000 ","End":"00:49.160","Text":"We\u0027re gonna use the theorem for that."},{"Start":"00:49.160 ","End":"00:55.595","Text":"I\u0027ll remind you, the conditions are that f should be continuous on the interval."},{"Start":"00:55.595 ","End":"00:58.595","Text":"It should be equal at the endpoints,"},{"Start":"00:58.595 ","End":"01:04.100","Text":"and it should be piecewise continuously differentiable."},{"Start":"01:04.100 ","End":"01:08.430","Text":"Then we can do term by term differentiation."},{"Start":"01:08.860 ","End":"01:11.600","Text":"The first part, continuous."},{"Start":"01:11.600 ","End":"01:14.120","Text":"We showed that in part a,"},{"Start":"01:14.120 ","End":"01:17.360","Text":"as for f(minus Pi)=f(Pi),"},{"Start":"01:17.360 ","End":"01:23.790","Text":"well in general, f on the whole line is 2Pi periodic."},{"Start":"01:23.790 ","End":"01:25.809","Text":"Well, it\u0027s just simple algebra."},{"Start":"01:25.809 ","End":"01:30.055","Text":"Basically it follows from the fact that each of the 2Pi i is 1."},{"Start":"01:30.055 ","End":"01:33.160","Text":"In particular, f(minus Pi) is f(Pi)."},{"Start":"01:33.160 ","End":"01:38.060","Text":"Now, suppose on the contrary that f is continuously differentiable,"},{"Start":"01:38.060 ","End":"01:42.195","Text":"then we have all 3 conditions for the theorem."},{"Start":"01:42.195 ","End":"01:48.010","Text":"Then we can differentiate term by term on this interval minus Pi Pi."},{"Start":"01:48.010 ","End":"01:50.680","Text":"When we differentiate each term,"},{"Start":"01:50.680 ","End":"01:54.610","Text":"we get the in^2 from the exponent here."},{"Start":"01:54.610 ","End":"01:56.770","Text":"Everything else stays the same."},{"Start":"01:56.770 ","End":"01:59.650","Text":"Now we\u0027ll apply the Riemann Lebesgue lemma,"},{"Start":"01:59.650 ","End":"02:06.620","Text":"which says that the general coefficient turns to 0 as n goes to infinity."},{"Start":"02:06.620 ","End":"02:08.400","Text":"Limit as n goes to infinity,"},{"Start":"02:08.400 ","End":"02:12.320","Text":"in^2 over 1 plus n^2=0, by this lemma."},{"Start":"02:12.320 ","End":"02:15.920","Text":"On the other hand, if we actually check what the limit is,"},{"Start":"02:15.920 ","End":"02:18.905","Text":"n^2 over 1 plus n^2 turns to 1."},{"Start":"02:18.905 ","End":"02:22.534","Text":"This thing turns to I, which is not 0."},{"Start":"02:22.534 ","End":"02:25.375","Text":"This and this gives us a contradiction."},{"Start":"02:25.375 ","End":"02:31.100","Text":"The contradiction came from assuming that f is continuously differentiable."},{"Start":"02:31.100 ","End":"02:34.115","Text":"F is not continuously differentiable."},{"Start":"02:34.115 ","End":"02:36.570","Text":"That concludes part b."}],"ID":28780},{"Watched":false,"Name":"Exercise 9 - Part a","Duration":"2m 6s","ChapterTopicVideoID":27617,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.155","Text":"In this exercise, f is a real-valued function on all of the real line,"},{"Start":"00:05.155 ","End":"00:07.900","Text":"and it\u0027s given by the following series."},{"Start":"00:07.900 ","End":"00:13.550","Text":"We\u0027ll show during the course of proving a that this actually does converge."},{"Start":"00:13.550 ","End":"00:16.755","Text":"We have to prove that f is continuous,"},{"Start":"00:16.755 ","End":"00:20.920","Text":"and in part b, that f is not continuously differentiable."},{"Start":"00:20.920 ","End":"00:23.110","Text":"Let\u0027s start with part a."},{"Start":"00:23.110 ","End":"00:26.995","Text":"We\u0027ll take each term and define it as f_n(x),"},{"Start":"00:26.995 ","End":"00:29.830","Text":"so that f is the sum of f_n."},{"Start":"00:29.830 ","End":"00:32.440","Text":"Each of the f_n is clearly continuous,"},{"Start":"00:32.440 ","End":"00:35.650","Text":"and if we show that the series converges uniformly,"},{"Start":"00:35.650 ","End":"00:39.720","Text":"then the conclusion is that f is also continuous."},{"Start":"00:39.720 ","End":"00:43.700","Text":"We\u0027re using the following theorem that we\u0027ve used before,"},{"Start":"00:43.700 ","End":"00:50.080","Text":"basically says that the uniform sum of series of continuous functions is also continuous."},{"Start":"00:50.080 ","End":"00:54.990","Text":"We\u0027ll use the Weierstrass M-test to show uniform convergence."},{"Start":"00:54.990 ","End":"00:58.490","Text":"Here is the Weierstrass M-test for reference."},{"Start":"00:58.490 ","End":"01:02.120","Text":"We have to find the sequence M_n,"},{"Start":"01:02.120 ","End":"01:07.870","Text":"which is convergent and which is an upper bound for each of the f_n."},{"Start":"01:07.870 ","End":"01:13.010","Text":"Let\u0027s see, the absolute value of f_n(x) is equal to the absolute value of this."},{"Start":"01:13.010 ","End":"01:16.235","Text":"The sign is less than or equal to 1,"},{"Start":"01:16.235 ","End":"01:20.800","Text":"and this is positive so we can say this is less than or equal to this,"},{"Start":"01:20.800 ","End":"01:24.100","Text":"and we\u0027ll let this be M_n."},{"Start":"01:24.100 ","End":"01:29.610","Text":"We just have to show now that the sum of M_n is finite,"},{"Start":"01:29.610 ","End":"01:32.009","Text":"that M_n is a convergent series."},{"Start":"01:32.009 ","End":"01:38.925","Text":"The sum of M_n is less than or equal to, we can replace n^3 plus 1 by n^3,"},{"Start":"01:38.925 ","End":"01:40.440","Text":"make the denominator smaller,"},{"Start":"01:40.440 ","End":"01:41.835","Text":"it only can get bigger."},{"Start":"01:41.835 ","End":"01:43.635","Text":"This is n^1/2,"},{"Start":"01:43.635 ","End":"01:45.180","Text":"this is n^3,"},{"Start":"01:45.180 ","End":"01:48.220","Text":"so this is n^2 and 1/2."},{"Start":"01:48.220 ","End":"01:54.830","Text":"This series converges because it\u0027s a p-series with p=2.5,"},{"Start":"01:54.830 ","End":"01:57.505","Text":"which is bigger than 1."},{"Start":"01:57.505 ","End":"01:59.595","Text":"By the M-test,"},{"Start":"01:59.595 ","End":"02:02.190","Text":"this converges uniformly,"},{"Start":"02:02.190 ","End":"02:04.970","Text":"so f is continuous and that\u0027s what we had to show,"},{"Start":"02:04.970 ","End":"02:07.260","Text":"and that concludes part a."}],"ID":28781},{"Watched":false,"Name":"Exercise 9 - Part b","Duration":"3m 32s","ChapterTopicVideoID":27618,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"Now we come to part b,"},{"Start":"00:02.730 ","End":"00:04.890","Text":"we just proved that f is continuous."},{"Start":"00:04.890 ","End":"00:08.775","Text":"Now we\u0027re going to prove that it\u0027s not continuously differentiable."},{"Start":"00:08.775 ","End":"00:13.185","Text":"Let\u0027s consider f just on the interval from minus Pi to Pi."},{"Start":"00:13.185 ","End":"00:16.050","Text":"Because f is equal to the sum,"},{"Start":"00:16.050 ","End":"00:20.450","Text":"then it\u0027s also represented by this Fourier series,"},{"Start":"00:20.450 ","End":"00:24.590","Text":"a real Fourier series with just sines and no cosines."},{"Start":"00:24.590 ","End":"00:29.855","Text":"We\u0027re getting ready to use the theorem on term-by-term differentiation."},{"Start":"00:29.855 ","End":"00:34.160","Text":"This is the complex version but it also works for the real version."},{"Start":"00:34.160 ","End":"00:36.920","Text":"We have a function f(minus Pi,"},{"Start":"00:36.920 ","End":"00:40.155","Text":"Pi) and there are 3 things, it\u0027s continuous,"},{"Start":"00:40.155 ","End":"00:43.010","Text":"it\u0027s equal at the endpoints,"},{"Start":"00:43.010 ","End":"00:47.060","Text":"and it\u0027s piecewise continuously differentiable."},{"Start":"00:47.060 ","End":"00:51.710","Text":"In this case, we can use term-by-term differentiation."},{"Start":"00:51.710 ","End":"00:57.065","Text":"Like I said, it also works for the real Fourier expansion."},{"Start":"00:57.065 ","End":"00:59.040","Text":"Back here, in part a,"},{"Start":"00:59.040 ","End":"01:01.485","Text":"we showed that f is continuous."},{"Start":"01:01.485 ","End":"01:07.950","Text":"Also, we can check that f( minus Pi) is f(Pi) because at minus Pi it\u0027s"},{"Start":"01:07.950 ","End":"01:15.410","Text":"0 and at Pi it is 0 because sine of some integer times Pi or minus Pi is 0."},{"Start":"01:15.410 ","End":"01:17.510","Text":"The third thing, well,"},{"Start":"01:17.510 ","End":"01:18.830","Text":"for this, we need to,"},{"Start":"01:18.830 ","End":"01:22.600","Text":"suppose on the contrary that f is continuously differentiable."},{"Start":"01:22.600 ","End":"01:24.500","Text":"The idea is to reach a contradiction."},{"Start":"01:24.500 ","End":"01:26.310","Text":"If it does satisfy this,"},{"Start":"01:26.310 ","End":"01:28.635","Text":"then we have all 3 conditions."},{"Start":"01:28.635 ","End":"01:30.900","Text":"Then on the interval minus Pi,"},{"Start":"01:30.900 ","End":"01:33.120","Text":"Pi we have that f satisfies"},{"Start":"01:33.120 ","End":"01:37.110","Text":"the 3 conditions and so we can do term-by-term differentiation."},{"Start":"01:37.110 ","End":"01:40.650","Text":"F\u0027(x) has a Fourier series."},{"Start":"01:40.650 ","End":"01:43.385","Text":"Well, term-by-term differentiation of this,"},{"Start":"01:43.385 ","End":"01:49.290","Text":"we just take an extra n^2 and put it here and the sine becomes a cosine."},{"Start":"01:49.290 ","End":"01:52.545","Text":"We have n^2.5,"},{"Start":"01:52.545 ","End":"01:54.240","Text":"and here, yeah, cosine,"},{"Start":"01:54.240 ","End":"01:56.445","Text":"and everything else is the same."},{"Start":"01:56.445 ","End":"01:59.105","Text":"In the previous exercise,"},{"Start":"01:59.105 ","End":"02:03.470","Text":"we use the Riemann-Lebesgue lemma but it\u0027s not"},{"Start":"02:03.470 ","End":"02:08.355","Text":"going to work here so we\u0027re going to use Parseval\u0027s identity."},{"Start":"02:08.355 ","End":"02:13.375","Text":"Parseval\u0027s identity in the real case says that 1 over Pi,"},{"Start":"02:13.375 ","End":"02:16.760","Text":"the integral of the absolute value of (f\u0027)^2 is"},{"Start":"02:16.760 ","End":"02:22.400","Text":"the sum of all the coefficients and absolute value squared."},{"Start":"02:22.400 ","End":"02:28.155","Text":"These are the coefficients that are non-zero, absolute value squared."},{"Start":"02:28.155 ","End":"02:30.758","Text":"Let\u0027s see what this is equal to."},{"Start":"02:30.758 ","End":"02:35.120","Text":"N^2.5^2 is n^5."},{"Start":"02:35.120 ","End":"02:42.650","Text":"Then we have n^3 plus 1^2 and we can estimate this as bigger or equal to the following."},{"Start":"02:42.650 ","End":"02:46.650","Text":"Replace this 1 by n^3 so making the denominator"},{"Start":"02:46.650 ","End":"02:51.810","Text":"bigger so the fraction is smaller so this is bigger or equal to."},{"Start":"02:51.810 ","End":"02:58.800","Text":"Then this is equal to n^3 plus n^3^2 is 4n^6."},{"Start":"02:58.800 ","End":"03:04.200","Text":"N^5 over 4n^6 is 1/4 1 over n. We"},{"Start":"03:04.200 ","End":"03:10.520","Text":"have 1/4 of the sum of the harmonic series which is known to be infinity."},{"Start":"03:10.520 ","End":"03:16.190","Text":"It diverges which is a contradiction because this integral is the"},{"Start":"03:16.190 ","End":"03:19.370","Text":"integral of a piecewise continuous function on"},{"Start":"03:19.370 ","End":"03:23.000","Text":"a closed interval so it has to be less than infinity."},{"Start":"03:23.000 ","End":"03:30.240","Text":"This is the contradiction which means that f is not continuously differentiable."},{"Start":"03:30.240 ","End":"03:32.860","Text":"That concludes part b."}],"ID":28782},{"Watched":false,"Name":"Exercise 10 - Part a","Duration":"1m 37s","ChapterTopicVideoID":27619,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, f is a real function on"},{"Start":"00:03.360 ","End":"00:07.290","Text":"the whole real line given by the following series,"},{"Start":"00:07.290 ","End":"00:09.915","Text":"which as we\u0027ll see, converges."},{"Start":"00:09.915 ","End":"00:13.065","Text":"We have to prove in Part a that f is continuous,"},{"Start":"00:13.065 ","End":"00:16.890","Text":"and in Part b that f is not continuously differentiable."},{"Start":"00:16.890 ","End":"00:22.485","Text":"We\u0027ll start with a and let this general term be f_n."},{"Start":"00:22.485 ","End":"00:25.200","Text":"F_n is the sum of these 2."},{"Start":"00:25.200 ","End":"00:29.040","Text":"Then f(x) will be the sum of the f_ns,"},{"Start":"00:29.040 ","End":"00:30.525","Text":"n goes 1 to infinity."},{"Start":"00:30.525 ","End":"00:33.615","Text":"Each of these f_ns is clearly continuous."},{"Start":"00:33.615 ","End":"00:37.115","Text":"If we show that the series converges uniformly,"},{"Start":"00:37.115 ","End":"00:39.830","Text":"then we can conclude that the sum f is also"},{"Start":"00:39.830 ","End":"00:43.910","Text":"continuous by the theorem that we\u0027ve seen before."},{"Start":"00:43.910 ","End":"00:46.645","Text":"Well, I\u0027ll just leave it here and you can read it."},{"Start":"00:46.645 ","End":"00:50.180","Text":"The way we will prove that the series is uniformly"},{"Start":"00:50.180 ","End":"00:53.720","Text":"convergent is by means of the Weierstrass M-test."},{"Start":"00:53.720 ","End":"00:55.520","Text":"I\u0027ll remind you of what that is."},{"Start":"00:55.520 ","End":"00:57.890","Text":"Again, I won\u0027t read it, it\u0027s here for reference."},{"Start":"00:57.890 ","End":"01:02.720","Text":"Let\u0027s see. Absolute value of f_n(x) is the absolute value of this."},{"Start":"01:02.720 ","End":"01:06.005","Text":"Then we can break it up using the triangle inequality."},{"Start":"01:06.005 ","End":"01:10.535","Text":"Now, cosine and sine are both an absolute value less than or equal to 1."},{"Start":"01:10.535 ","End":"01:14.385","Text":"We get this and we\u0027ll let this be M_n."},{"Start":"01:14.385 ","End":"01:17.790","Text":"Now the sum, we can split it up into 2 sums."},{"Start":"01:17.790 ","End":"01:23.195","Text":"Each of these is convergent because each of these is a p-series with p bigger than 1,"},{"Start":"01:23.195 ","End":"01:26.600","Text":"1.4 and 2.8 are both bigger than 1."},{"Start":"01:26.600 ","End":"01:32.270","Text":"By the M-test, the sum of f_n is f uniformly on R,"},{"Start":"01:32.270 ","End":"01:37.680","Text":"so f is continuous on all of the reals. That\u0027s part a."}],"ID":28783},{"Watched":false,"Name":"Exercise 10 - Part b","Duration":"3m 9s","ChapterTopicVideoID":27591,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"Now we come to Part B of this exercise."},{"Start":"00:03.270 ","End":"00:06.225","Text":"We\u0027ve just proven Part A that f is continuous."},{"Start":"00:06.225 ","End":"00:09.300","Text":"Now we\u0027re going to show that f is not continuously differentiable."},{"Start":"00:09.300 ","End":"00:13.139","Text":"Let\u0027s restrict f to the interval minus Pi, Pi."},{"Start":"00:13.139 ","End":"00:18.030","Text":"On this interval, f is represented by the Fourier series."},{"Start":"00:18.030 ","End":"00:22.680","Text":"Well, the same as this but just we write it as with a tilde."},{"Start":"00:22.680 ","End":"00:25.890","Text":"Because we can\u0027t do the other way around if f is represented"},{"Start":"00:25.890 ","End":"00:29.010","Text":"by the constant it\u0027s equal to but this way around we can say."},{"Start":"00:29.010 ","End":"00:34.875","Text":"Now we\u0027ve shown that f is continuous and also f is 2 Pi periodic."},{"Start":"00:34.875 ","End":"00:40.250","Text":"It\u0027s a simple calculation if you put x plus 2 Pi instead of x because"},{"Start":"00:40.250 ","End":"00:43.250","Text":"the cosine of something plus 2n Pi and"},{"Start":"00:43.250 ","End":"00:47.135","Text":"the sine of something plus 2n Pi is the sum of cosine and sine of that something,"},{"Start":"00:47.135 ","End":"00:50.165","Text":"we get that f of x plus 2 Pi is f of x."},{"Start":"00:50.165 ","End":"00:54.019","Text":"In particular, f of minus Pi is f of Pi"},{"Start":"00:54.019 ","End":"00:59.570","Text":". Now suppose on the contrary that f is continuously differentiable,"},{"Start":"00:59.570 ","End":"01:04.205","Text":"then f satisfies the conditions of the term by term differentiation theorem."},{"Start":"01:04.205 ","End":"01:05.960","Text":"Here is the theorem."},{"Start":"01:05.960 ","End":"01:09.439","Text":"This is for a complex function but it works also for a real function."},{"Start":"01:09.439 ","End":"01:16.190","Text":"We\u0027ve shown that f is continuous and that it\u0027s equal at"},{"Start":"01:16.190 ","End":"01:19.850","Text":"the end points and we\u0027re assuming by contradiction that"},{"Start":"01:19.850 ","End":"01:24.430","Text":"it\u0027s piecewise continuously differentiable."},{"Start":"01:24.430 ","End":"01:28.310","Text":"Then we can do term by term differentiation."},{"Start":"01:28.310 ","End":"01:33.965","Text":"We already said that it also works for real Fourier expansion."},{"Start":"01:33.965 ","End":"01:37.858","Text":"Back here. Term by term differentiation."},{"Start":"01:37.858 ","End":"01:43.625","Text":"This part here becomes this part here."},{"Start":"01:43.625 ","End":"01:48.290","Text":"The 2.8 becomes 1.8 because the derivative of sine"},{"Start":"01:48.290 ","End":"01:52.810","Text":"is cosine but we also have an extra n from the inner derivative."},{"Start":"01:52.810 ","End":"01:59.635","Text":"Similarly, this part here with the cosine becomes minus sine."},{"Start":"01:59.635 ","End":"02:04.820","Text":"Again, because the n comes out instead of 1.4, we have 0.4."},{"Start":"02:04.820 ","End":"02:07.130","Text":"That will apply Parseval\u0027s theorem to"},{"Start":"02:07.130 ","End":"02:13.580","Text":"this Fourier expansion and we get 1 over Pi the integral of absolute value of f\u0027^2"},{"Start":"02:13.580 ","End":"02:18.110","Text":"is the sum of the absolute value squared of"},{"Start":"02:18.110 ","End":"02:20.750","Text":"all the coefficients but not including"},{"Start":"02:20.750 ","End":"02:24.305","Text":"the cosine in x and the sine in x so it\u0027s just 1 over."},{"Start":"02:24.305 ","End":"02:27.350","Text":"Now, this is bigger or equal to,"},{"Start":"02:27.350 ","End":"02:31.160","Text":"I can just throw out the first term and the second"},{"Start":"02:31.160 ","End":"02:36.710","Text":"becomes 1 over n to the power of twice 0.4, which is 0.8."},{"Start":"02:36.710 ","End":"02:40.830","Text":"This is also a p-series but p is less than 1"},{"Start":"02:40.830 ","End":"02:46.145","Text":"here and so this diverges it goes to infinity."},{"Start":"02:46.145 ","End":"02:51.845","Text":"We get a contradiction because this integral is finite."},{"Start":"02:51.845 ","End":"02:55.370","Text":"It\u0027s piecewise continuous on a finite interval"},{"Start":"02:55.370 ","End":"02:59.245","Text":"so bounded on a finite interval so the integral is finite."},{"Start":"02:59.245 ","End":"03:01.140","Text":"On the other hand it\u0027s infinite."},{"Start":"03:01.140 ","End":"03:04.850","Text":"This contradiction came from supposing that f"},{"Start":"03:04.850 ","End":"03:10.140","Text":"is continuously differentiable and so it isn\u0027t and that concludes Part B."}],"ID":28784},{"Watched":false,"Name":"Decay Rate of Fourier Coefficients Theorem - Part 1","Duration":"6m 55s","ChapterTopicVideoID":27592,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.320 ","End":"00:03.360","Text":"We come to a new topic,"},{"Start":"00:03.360 ","End":"00:10.290","Text":"smoothness of a function and its relation to the decay rate of its Fourier coefficients."},{"Start":"00:10.290 ","End":"00:12.345","Text":"We\u0027ll start with a theorem,"},{"Start":"00:12.345 ","End":"00:15.060","Text":"there\u0027s a complex version and a real version,"},{"Start":"00:15.060 ","End":"00:17.605","Text":"and I\u0027ve combined them into one."},{"Start":"00:17.605 ","End":"00:22.330","Text":"Let f be a real or complex valued function on the real line,"},{"Start":"00:22.330 ","End":"00:25.075","Text":"and suppose it\u0027s 2 Pi periodic,"},{"Start":"00:25.075 ","End":"00:27.770","Text":"every 2 Pi, it repeats itself."},{"Start":"00:27.770 ","End":"00:30.335","Text":"Let k be a natural number."},{"Start":"00:30.335 ","End":"00:33.205","Text":"Natural number could include 0 or not,"},{"Start":"00:33.205 ","End":"00:34.845","Text":"it works both ways."},{"Start":"00:34.845 ","End":"00:39.515","Text":"Now, suppose that f is a function in C^k minus 1,"},{"Start":"00:39.515 ","End":"00:44.255","Text":"this means continuously differentiable k minus 1 times."},{"Start":"00:44.255 ","End":"00:51.600","Text":"F^k, the kth derivative of f is piecewise continuous on R."},{"Start":"00:51.600 ","End":"00:59.305","Text":"Then the limit as n goes to plus or minus infinity of n^k c_n is 0."},{"Start":"00:59.305 ","End":"01:02.390","Text":"If to see it written with absolute value,"},{"Start":"01:02.390 ","End":"01:05.465","Text":"but I guess it works without the absolute value also."},{"Start":"01:05.465 ","End":"01:07.670","Text":"In the real case,"},{"Start":"01:07.670 ","End":"01:15.180","Text":"we have two equalities n^k a_n for the coefficients of the cosine and n^k b_n,"},{"Start":"01:15.180 ","End":"01:16.885","Text":"the coefficients of the sine."},{"Start":"01:16.885 ","End":"01:19.505","Text":"These two limits are also 0."},{"Start":"01:19.505 ","End":"01:23.735","Text":"The c_n and the a_n are the Fourier coefficients."},{"Start":"01:23.735 ","End":"01:28.525","Text":"In the complex case this is where we get the c_n"},{"Start":"01:28.525 ","End":"01:34.665","Text":"and in the real case we have the a_n and the b_n."},{"Start":"01:34.665 ","End":"01:39.230","Text":"Now, we can phrase this in words informally and say"},{"Start":"01:39.230 ","End":"01:46.485","Text":"that c_n tends to 0 faster than 1 over n^k,"},{"Start":"01:46.485 ","End":"01:52.275","Text":"c_n divided by 1 over n^k is n^k c_n and this goes to 0."},{"Start":"01:52.275 ","End":"01:54.600","Text":"The ratio of these goes to 0,"},{"Start":"01:54.600 ","End":"01:57.090","Text":"so this certainly goes to 0 faster."},{"Start":"01:57.090 ","End":"02:02.445","Text":"Similarly a_n and b_n goes to 0 faster than 1 over n^k."},{"Start":"02:02.445 ","End":"02:05.780","Text":"Very informally, the smoother the function,"},{"Start":"02:05.780 ","End":"02:08.405","Text":"the quicker the coefficients tend to 0,"},{"Start":"02:08.405 ","End":"02:10.610","Text":"because the bigger k is,"},{"Start":"02:10.610 ","End":"02:12.545","Text":"the smoother f is."},{"Start":"02:12.545 ","End":"02:15.710","Text":"As k increases, it has more and more derivatives,"},{"Start":"02:15.710 ","End":"02:17.065","Text":"so it\u0027s smoother,"},{"Start":"02:17.065 ","End":"02:19.830","Text":"and as k gets bigger here,"},{"Start":"02:19.830 ","End":"02:23.520","Text":"this thing goes to 0 quicker and c_n even quicker,"},{"Start":"02:23.520 ","End":"02:24.860","Text":"so yeah, smoother the function,"},{"Start":"02:24.860 ","End":"02:27.349","Text":"the quicker the coefficients tend to 0."},{"Start":"02:27.349 ","End":"02:29.615","Text":"Now let\u0027s prove the theorem."},{"Start":"02:29.615 ","End":"02:33.980","Text":"This is not a difficult proof as long as you know proof by induction."},{"Start":"02:33.980 ","End":"02:37.565","Text":"Let\u0027s see what we know about f, some of the things."},{"Start":"02:37.565 ","End":"02:42.335","Text":"Well, it\u0027s representation of the Fourier series is this."},{"Start":"02:42.335 ","End":"02:44.345","Text":"We know it\u0027s continuous."},{"Start":"02:44.345 ","End":"02:47.960","Text":"We know f(Pi) is f (-Pi)."},{"Start":"02:47.960 ","End":"02:54.065","Text":"We know that it\u0027s piecewise continuously differentiable on the interval minus Pi, Pi."},{"Start":"02:54.065 ","End":"02:58.160","Text":"From all this, we can conclude that f"},{"Start":"02:58.160 ","End":"03:03.225","Text":"satisfies the conditions of the term by term differentiation theorem,"},{"Start":"03:03.225 ","End":"03:09.110","Text":"so differentiate term by term and we get f\u0027 is represented by same thing almost,"},{"Start":"03:09.110 ","End":"03:11.270","Text":"except with the in, in front,"},{"Start":"03:11.270 ","End":"03:14.375","Text":"which we get by differentiating e^inx."},{"Start":"03:14.375 ","End":"03:18.755","Text":"You could say that each time we differentiate we get an extra in,"},{"Start":"03:18.755 ","End":"03:27.520","Text":"so we could say that the kth derivative is the sum of in to the k, c_n e^inx."},{"Start":"03:27.520 ","End":"03:28.970","Text":"We can say it informally,"},{"Start":"03:28.970 ","End":"03:32.045","Text":"but we need to prove it and we\u0027ll prove it by induction."},{"Start":"03:32.045 ","End":"03:34.775","Text":"We just did the case where k=1,"},{"Start":"03:34.775 ","End":"03:36.205","Text":"which is written here."},{"Start":"03:36.205 ","End":"03:41.990","Text":"In fact, it\u0027s also true for k=0 in case you want the natural numbers to start at 0."},{"Start":"03:41.990 ","End":"03:49.110","Text":"When k=0, we just get the original representation of f as a Fourier series."},{"Start":"03:49.110 ","End":"03:50.915","Text":"We have k=0, k=1."},{"Start":"03:50.915 ","End":"03:53.285","Text":"Now we want to do an induction step."},{"Start":"03:53.285 ","End":"03:56.810","Text":"We\u0027re going to assume it\u0027s true for a particular k,"},{"Start":"03:56.810 ","End":"04:02.185","Text":"and we\u0027ll prove that it holds also if you replace k by k plus 1."},{"Start":"04:02.185 ","End":"04:04.615","Text":"What do we have to do with k plus 1?"},{"Start":"04:04.615 ","End":"04:09.485","Text":"This time we\u0027re given a function f instead of C^k minus 1 in C^k."},{"Start":"04:09.485 ","End":"04:12.445","Text":"Instead of f^k we have f^k plus 1."},{"Start":"04:12.445 ","End":"04:14.085","Text":"Piecewise continuous."},{"Start":"04:14.085 ","End":"04:17.720","Text":"We have to show again what we wanted to show except"},{"Start":"04:17.720 ","End":"04:21.725","Text":"we replace k by k plus 1 here and here."},{"Start":"04:21.725 ","End":"04:23.560","Text":"Now here\u0027s the trick,"},{"Start":"04:23.560 ","End":"04:26.160","Text":"let big F be F\u0027."},{"Start":"04:26.160 ","End":"04:28.940","Text":"If you formulate this all in terms of big F,"},{"Start":"04:28.940 ","End":"04:31.505","Text":"the indexes will go down by 1,"},{"Start":"04:31.505 ","End":"04:35.315","Text":"because look, f belongs to C^k minus 1."},{"Start":"04:35.315 ","End":"04:39.470","Text":"I mean, if little f is differentiable k times then"},{"Start":"04:39.470 ","End":"04:43.820","Text":"the first derivative is differentiable k minus 1 times."},{"Start":"04:43.820 ","End":"04:49.745","Text":"The k plus 1 derivative of f is just the kth derivative of big F. Now,"},{"Start":"04:49.745 ","End":"04:51.560","Text":"to copy something we wrote already,"},{"Start":"04:51.560 ","End":"04:54.275","Text":"f\u0027(x) is this series,"},{"Start":"04:54.275 ","End":"04:58.535","Text":"and that means if you replace f\u0027 by big F,"},{"Start":"04:58.535 ","End":"05:04.745","Text":"and we replace this inc_n by a different name,"},{"Start":"05:04.745 ","End":"05:10.880","Text":"big C_n, this is the Fourier representation of big F. Now we have"},{"Start":"05:10.880 ","End":"05:16.920","Text":"the representation of f and it satisfies all the conditions of the original little f,"},{"Start":"05:16.920 ","End":"05:21.285","Text":"belongs to C^k minus 1, piecewise continuous here,"},{"Start":"05:21.285 ","End":"05:27.740","Text":"so we can just apply the induction hypothesis but replace f by big F. We"},{"Start":"05:27.740 ","End":"05:34.745","Text":"have the kth derivative of big F is represented by in^k, C_n, e^inx."},{"Start":"05:34.745 ","End":"05:38.105","Text":"Now all we have to do is convert this back to little f,"},{"Start":"05:38.105 ","End":"05:41.660","Text":"and this is kth derivative of f\u0027."},{"Start":"05:41.660 ","End":"05:43.925","Text":"Here we get in^k,"},{"Start":"05:43.925 ","End":"05:45.560","Text":"and this is inC_n,"},{"Start":"05:45.560 ","End":"05:55.250","Text":"so it just simplifies to f^k plus 1 derivative is represented by in^k plus 1 C_n e^inx."},{"Start":"05:55.250 ","End":"05:57.230","Text":"I don\u0027t know why I copied this again,"},{"Start":"05:57.230 ","End":"05:59.240","Text":"but yeah, this is what we have to show."},{"Start":"05:59.240 ","End":"06:01.220","Text":"This is the case k plus 1."},{"Start":"06:01.220 ","End":"06:05.185","Text":"We assume it\u0027s true for k and now we\u0027ve shown is true for k plus 1."},{"Start":"06:05.185 ","End":"06:07.865","Text":"Just to repeat the claim with k again,"},{"Start":"06:07.865 ","End":"06:09.529","Text":"this is what we\u0027ve proved."},{"Start":"06:09.529 ","End":"06:13.090","Text":"We showed it\u0027s true for k=0 and 1 in fact,"},{"Start":"06:13.090 ","End":"06:15.270","Text":"and we also showed the induction step,"},{"Start":"06:15.270 ","End":"06:17.255","Text":"so yeah, we\u0027ve done this by induction."},{"Start":"06:17.255 ","End":"06:20.090","Text":"Now it\u0027s easy to finish off the proof."},{"Start":"06:20.090 ","End":"06:21.410","Text":"Now that we have this,"},{"Start":"06:21.410 ","End":"06:26.165","Text":"we can use the Riemann–Lebesgue lemma that the coefficient"},{"Start":"06:26.165 ","End":"06:32.045","Text":"goes to 0 as n goes to plus or minus infinity."},{"Start":"06:32.045 ","End":"06:36.125","Text":"Now we can take i^k outside the brackets,"},{"Start":"06:36.125 ","End":"06:38.540","Text":"and the absolute value of i^k is 1,"},{"Start":"06:38.540 ","End":"06:40.430","Text":"but even if it isn\u0027t 1 is non-zero,"},{"Start":"06:40.430 ","End":"06:45.200","Text":"so we can divide both sides and just get left with this part."},{"Start":"06:45.200 ","End":"06:50.585","Text":"That the limit as n goes to plus or minus infinity of n^k C_n is 0."},{"Start":"06:50.585 ","End":"06:52.685","Text":"That\u0027s what we had to prove,"},{"Start":"06:52.685 ","End":"06:55.950","Text":"and that concludes this exercise."}],"ID":28785},{"Watched":false,"Name":"Decay Rate of Fourier Coefficients Theorem - Part 2","Duration":"1m 45s","ChapterTopicVideoID":27593,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"We\u0027re continuing with the relation between the smoothness of"},{"Start":"00:03.570 ","End":"00:06.975","Text":"a function and the decay rate of its Fourier coefficients."},{"Start":"00:06.975 ","End":"00:10.109","Text":"We said earlier that the smoother the function,"},{"Start":"00:10.109 ","End":"00:13.170","Text":"the quicker the coefficients tend to 0."},{"Start":"00:13.170 ","End":"00:18.840","Text":"There is also an inverse rule that the quicker the coefficients tend to 0,"},{"Start":"00:18.840 ","End":"00:20.415","Text":"the smoother the function."},{"Start":"00:20.415 ","End":"00:22.260","Text":"We can state this in a theorem."},{"Start":"00:22.260 ","End":"00:23.580","Text":"This is the second theorem."},{"Start":"00:23.580 ","End":"00:26.405","Text":"The first theorem was for the one direction,"},{"Start":"00:26.405 ","End":"00:29.230","Text":"now we have a theorem for the other direction."},{"Start":"00:29.450 ","End":"00:32.310","Text":"We\u0027ll take the complex case."},{"Start":"00:32.310 ","End":"00:34.470","Text":"Let f be a complex valued function,"},{"Start":"00:34.470 ","End":"00:37.735","Text":"2Pi periodic and piecewise continuous."},{"Start":"00:37.735 ","End":"00:41.510","Text":"Let\u0027s say that this is its Fourier representation."},{"Start":"00:41.510 ","End":"00:49.340","Text":"Now, suppose that there is an a natural number k and real numbers c and Epsilon"},{"Start":"00:49.340 ","End":"00:53.190","Text":"such that absolute value of c_n is less than or equal"},{"Start":"00:53.190 ","End":"00:58.910","Text":"to c over n^k plus 1 plus Epsilon for all integers except 0,"},{"Start":"00:58.910 ","End":"01:01.195","Text":"we don\u0027t want to divide by 0."},{"Start":"01:01.195 ","End":"01:05.470","Text":"Then f belongs to the class C^k,"},{"Start":"01:05.470 ","End":"01:09.769","Text":"meaning functions which are k times continuously differentiable."},{"Start":"01:09.769 ","End":"01:13.271","Text":"For the variation,"},{"Start":"01:13.271 ","End":"01:15.912","Text":"in some books or some sites,"},{"Start":"01:15.912 ","End":"01:21.320","Text":"instead of 1 plus Epsilon we have Alpha and instead of Epsilon bigger than 0,"},{"Start":"01:21.320 ","End":"01:24.680","Text":"Alpha bigger than 1, just a variation."},{"Start":"01:24.680 ","End":"01:26.900","Text":"We won\u0027t prove the theorem this time."},{"Start":"01:26.900 ","End":"01:30.350","Text":"Let\u0027s just state that the analogous theorem for real functions,"},{"Start":"01:30.350 ","End":"01:34.140","Text":"instead of having C_n less than we have a_n and"},{"Start":"01:34.140 ","End":"01:38.820","Text":"b_n both less than some constant over this."},{"Start":"01:38.820 ","End":"01:44.285","Text":"We\u0027ll use this reverse theorem in one of the exercises that follow."},{"Start":"01:44.285 ","End":"01:46.770","Text":"That\u0027s all."}],"ID":28786},{"Watched":false,"Name":"Exercise 11","Duration":"2m 29s","ChapterTopicVideoID":27594,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"In this exercise, f(x) is the sum of this series."},{"Start":"00:05.070 ","End":"00:07.605","Text":"This is in fact the Fourier series."},{"Start":"00:07.605 ","End":"00:13.860","Text":"We have to prove that f does not belong to the space C^4."},{"Start":"00:13.860 ","End":"00:18.130","Text":"In other words, f isn\u0027t continuously differentiable 4 times."},{"Start":"00:18.130 ","End":"00:20.475","Text":"We\u0027ll prove this by contradiction."},{"Start":"00:20.475 ","End":"00:25.050","Text":"We suppose on the contrary that f does belong to C^4,"},{"Start":"00:25.050 ","End":"00:28.080","Text":"and get a contradiction eventually."},{"Start":"00:28.080 ","End":"00:33.270","Text":"Let\u0027s remember the theorem that we had that if we have f which"},{"Start":"00:33.270 ","End":"00:38.493","Text":"is 2Pi periodic and k is a natural number."},{"Start":"00:38.493 ","End":"00:44.917","Text":"Let\u0027s say that f is represented by the following real Fourier series,"},{"Start":"00:44.917 ","End":"00:49.025","Text":"and we suppose that f belongs to C^k - 1,"},{"Start":"00:49.025 ","End":"00:52.460","Text":"and the kth derivative of f is piecewise continuous."},{"Start":"00:52.460 ","End":"00:58.160","Text":"We have all this. Then the limit as n goes to infinity of"},{"Start":"00:58.160 ","End":"01:05.555","Text":"n^k a_n is 0 and lim n goes to infinity of n^k b_n is 0."},{"Start":"01:05.555 ","End":"01:10.730","Text":"Well, in this case, we only have the b_n the a\u0027s are all 0,"},{"Start":"01:10.730 ","End":"01:13.630","Text":"so we just need to use this."},{"Start":"01:13.630 ","End":"01:16.880","Text":"Let\u0027s choose k=4."},{"Start":"01:16.880 ","End":"01:21.575","Text":"F belongs to C^4 by the assumption,"},{"Start":"01:21.575 ","End":"01:24.320","Text":"and therefore f belongs to C^3,"},{"Start":"01:24.320 ","End":"01:27.110","Text":"which is C^k minus 1."},{"Start":"01:27.110 ","End":"01:32.495","Text":"Also, the fourth derivative of f is continuous,"},{"Start":"01:32.495 ","End":"01:35.975","Text":"which means in particular that is piecewise continuous."},{"Start":"01:35.975 ","End":"01:40.100","Text":"Now we have the conditions of the theorem."},{"Start":"01:40.100 ","End":"01:43.115","Text":"This condition implies this and this,"},{"Start":"01:43.115 ","End":"01:46.580","Text":"which is this with k=4."},{"Start":"01:46.580 ","End":"01:48.770","Text":"Now, this is just what I just said,"},{"Start":"01:48.770 ","End":"01:56.395","Text":"that this series means that the a_n\u0027s are all 0 and the b_n\u0027s 1^4."},{"Start":"01:56.395 ","End":"02:00.320","Text":"If we apply the conclusion of this theorem,"},{"Start":"02:00.320 ","End":"02:03.980","Text":"we get that the limit n^k b_n is 0,"},{"Start":"02:03.980 ","End":"02:11.340","Text":"we\u0027re not cases translates to the limit of n^4 times 1^4 is 0."},{"Start":"02:11.340 ","End":"02:16.230","Text":"But the limit of this is the limit of 1, which is 1."},{"Start":"02:16.230 ","End":"02:18.105","Text":"It gives us 1=0,"},{"Start":"02:18.105 ","End":"02:20.565","Text":"which is certainly a contradiction."},{"Start":"02:20.565 ","End":"02:26.525","Text":"The contradiction came from assuming that f does belong to C^4, therefore it doesn\u0027t."},{"Start":"02:26.525 ","End":"02:30.120","Text":"That concludes this exercise."}],"ID":28787},{"Watched":false,"Name":"Exercise 12","Duration":"2m 33s","ChapterTopicVideoID":27595,"CourseChapterTopicPlaylistID":294456,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.780","Text":"In this exercise, f (x) is given as the sum of this series,"},{"Start":"00:06.780 ","End":"00:10.815","Text":"which has the form of a complex Fourier series."},{"Start":"00:10.815 ","End":"00:15.300","Text":"We have to prove that f belongs to C^2."},{"Start":"00:15.300 ","End":"00:19.755","Text":"In other words, f is twice continuously differentiable."},{"Start":"00:19.755 ","End":"00:22.908","Text":"We\u0027re going to use the theorem,"},{"Start":"00:22.908 ","End":"00:24.210","Text":"f is a complex function,"},{"Start":"00:24.210 ","End":"00:27.000","Text":"2 Pi periodic and piecewise continuous."},{"Start":"00:27.000 ","End":"00:30.585","Text":"This is its Fourier series representation,"},{"Start":"00:30.585 ","End":"00:35.420","Text":"and then we have a natural number k and positive c and"},{"Start":"00:35.420 ","End":"00:41.885","Text":"Epsilon such that this inequality holds for all n except 0."},{"Start":"00:41.885 ","End":"00:45.980","Text":"Then the conclusion is that f belongs to C^k."},{"Start":"00:45.980 ","End":"00:49.085","Text":"In other words, f is k times continuously differentiable."},{"Start":"00:49.085 ","End":"00:51.365","Text":"This is the theorem we\u0027ll be using,"},{"Start":"00:51.365 ","End":"00:54.830","Text":"the absolute value of C_n or let\u0027s just go back and look here."},{"Start":"00:54.830 ","End":"00:59.090","Text":"Here it is. This is C_n, the coefficient part."},{"Start":"00:59.090 ","End":"01:04.870","Text":"The absolute value of C_n is absolute value of this."},{"Start":"01:08.950 ","End":"01:15.140","Text":"The reason n is bigger or equal to 1 is because the series is defined from 1 to infinity."},{"Start":"01:15.140 ","End":"01:17.120","Text":"We don\u0027t have 0 or negative."},{"Start":"01:17.120 ","End":"01:21.150","Text":"In fact, they\u0027re all 0 back here."},{"Start":"01:21.150 ","End":"01:22.670","Text":"This is equal to,"},{"Start":"01:22.670 ","End":"01:25.385","Text":"and this is a complex number on the denominator,"},{"Start":"01:25.385 ","End":"01:27.220","Text":"a plus ib,"},{"Start":"01:27.220 ","End":"01:31.965","Text":"so the norm is the square root of a^2 plus b^2,"},{"Start":"01:31.965 ","End":"01:35.230","Text":"and we can just throw this 1 out."},{"Start":"01:35.230 ","End":"01:36.440","Text":"If we throw this out,"},{"Start":"01:36.440 ","End":"01:38.825","Text":"the denominator gets smaller,"},{"Start":"01:38.825 ","End":"01:42.720","Text":"so the fraction gets bigger."},{"Start":"01:42.720 ","End":"01:47.040","Text":"This is less than or equal to 2 over the square root of"},{"Start":"01:47.040 ","End":"01:51.700","Text":"a^2 is just a in absolute value, which is this."},{"Start":"01:51.700 ","End":"01:53.340","Text":"This is equal to,"},{"Start":"01:53.340 ","End":"02:01.845","Text":"we can split the 3.1 up into 2 plus 1 plus 0.1 to make it like this."},{"Start":"02:01.845 ","End":"02:06.080","Text":"K is 2 and Epsilon is 0.1,"},{"Start":"02:06.080 ","End":"02:09.365","Text":"and the C on the top is like the 2."},{"Start":"02:09.365 ","End":"02:11.060","Text":"We choose k equals 2,"},{"Start":"02:11.060 ","End":"02:13.100","Text":"c equals 2, Epsilon equals 0.1."},{"Start":"02:13.100 ","End":"02:16.610","Text":"Then we have the absolute value of C_n is less than"},{"Start":"02:16.610 ","End":"02:20.575","Text":"or equal to c over n to the k plus 1 plus Epsilon."},{"Start":"02:20.575 ","End":"02:26.360","Text":"Then we can apply the theorem and say that f belongs to C^k,"},{"Start":"02:26.360 ","End":"02:28.985","Text":"which in this case is C^2."},{"Start":"02:28.985 ","End":"02:34.140","Text":"That concludes the proof and this exercise."}],"ID":28788}],"Thumbnail":null,"ID":294456},{"Name":"Uniform Convergence of Fourier Series","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Uniform Convergence of Fourier Series","Duration":"3m 48s","ChapterTopicVideoID":27550,"CourseChapterTopicPlaylistID":294457,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.505","Text":"This clip is about uniform convergence of Fourier series, and particularly,"},{"Start":"00:05.505 ","End":"00:08.880","Text":"what are the conditions under which we can say for definite that"},{"Start":"00:08.880 ","End":"00:14.460","Text":"the Fourier series converges uniformly to the function that it represents."},{"Start":"00:14.460 ","End":"00:18.585","Text":"We\u0027ll start with the proposition theorem or lemma doesn\u0027t matter."},{"Start":"00:18.585 ","End":"00:26.340","Text":"If the function f and f\u0027 are both piecewise continuous on minus Pi, Pi,"},{"Start":"00:26.340 ","End":"00:29.900","Text":"and suppose that f is actually continuous,"},{"Start":"00:29.900 ","End":"00:32.765","Text":"not just piecewise on minus Pi,"},{"Start":"00:32.765 ","End":"00:35.840","Text":"Pi and f is equal at both endpoints,"},{"Start":"00:35.840 ","End":"00:38.165","Text":"f of minus Pi equals f(Pi)."},{"Start":"00:38.165 ","End":"00:39.950","Text":"Another way to say that is that"},{"Start":"00:39.950 ","End":"00:46.235","Text":"the 2Pi periodic extension f tilde is continuous on all of the reals."},{"Start":"00:46.235 ","End":"00:52.453","Text":"It\u0027s continuous on each interval of length 2Pi and if it matches at the end points."},{"Start":"00:52.453 ","End":"00:55.880","Text":"We can just keep duplicating it to the left and"},{"Start":"00:55.880 ","End":"00:59.645","Text":"to the right and get a function that\u0027s continuous on all of the reals."},{"Start":"00:59.645 ","End":"01:01.370","Text":"Well, in this case,"},{"Start":"01:01.370 ","End":"01:09.305","Text":"the Fourier series for f converges uniformly to the function f on the interval."},{"Start":"01:09.305 ","End":"01:18.935","Text":"In fact, it converges on all of our because of this condition and the periodicity."},{"Start":"01:18.935 ","End":"01:26.630","Text":"Note the similarity of the conditions on f to those in the term by term differentiation."},{"Start":"01:26.630 ","End":"01:29.690","Text":"They are related, but it won\u0027t say anymore than that."},{"Start":"01:29.690 ","End":"01:35.690","Text":"Now, what would happen if f(Pi) did not equal f(-Pi)?"},{"Start":"01:35.690 ","End":"01:40.655","Text":"That would be a discontinuity at minus Pi and Pi."},{"Start":"01:40.655 ","End":"01:43.940","Text":"Maybe we can still have uniform convergence,"},{"Start":"01:43.940 ","End":"01:47.450","Text":"but possibly on a different interval, a smaller interval."},{"Start":"01:47.450 ","End":"01:49.159","Text":"The answer is yes."},{"Start":"01:49.159 ","End":"01:55.129","Text":"If we just have the condition that f and f’ are piecewise continuous,"},{"Start":"01:55.129 ","End":"01:57.575","Text":"but without the condition on the endpoints,"},{"Start":"01:57.575 ","End":"02:02.630","Text":"then the Fourier series of f converges uniformly on"},{"Start":"02:02.630 ","End":"02:08.735","Text":"any interval ab on which f tilde is continuous,"},{"Start":"02:08.735 ","End":"02:14.180","Text":"meaning that it misses all the discontinuity points of the piecewise."},{"Start":"02:14.180 ","End":"02:17.225","Text":"Find an interval in which it is continuous,"},{"Start":"02:17.225 ","End":"02:20.855","Text":"then the Fourier series converges uniformly to it."},{"Start":"02:20.855 ","End":"02:22.790","Text":"That\u0027s it for the theory."},{"Start":"02:22.790 ","End":"02:25.250","Text":"Now let\u0027s do an example problem."},{"Start":"02:25.250 ","End":"02:29.325","Text":"We\u0027re given the function f(x) as follows,"},{"Start":"02:29.325 ","End":"02:31.815","Text":"where a and b are parameters,"},{"Start":"02:31.815 ","End":"02:34.690","Text":"it\u0027s defined on minus Pi to Pi."},{"Start":"02:34.690 ","End":"02:38.324","Text":"The question is for which values of the parameter a and b does"},{"Start":"02:38.324 ","End":"02:44.943","Text":"the Fourier series of f converge uniformly on all of the interval minus Pi?"},{"Start":"02:44.943 ","End":"02:53.090","Text":"What we\u0027d like is for f to be continuous and equal on the endpoints."},{"Start":"02:53.090 ","End":"02:56.285","Text":"For the continuity at zero,"},{"Start":"02:56.285 ","End":"03:01.730","Text":"we need for these to be equal when x equals zero,"},{"Start":"03:01.730 ","End":"03:02.795","Text":"put it another way,"},{"Start":"03:02.795 ","End":"03:08.809","Text":"the limit of f at zero from the left has got to equal to the limit on the right."},{"Start":"03:08.809 ","End":"03:12.630","Text":"Just substitute here x equals zero,"},{"Start":"03:12.630 ","End":"03:19.815","Text":"a substitute here x equals zero and compare and that will give us that b equals three."},{"Start":"03:19.815 ","End":"03:22.050","Text":"Now at the end points,"},{"Start":"03:22.050 ","End":"03:28.260","Text":"we want f(-Pi) to equal f(Pi)."},{"Start":"03:28.260 ","End":"03:34.040","Text":"Just plugging in substituting Pi here and minus Pi here,"},{"Start":"03:34.040 ","End":"03:35.900","Text":"we get this equation."},{"Start":"03:35.900 ","End":"03:38.420","Text":"We already know that b equals three,"},{"Start":"03:38.420 ","End":"03:41.750","Text":"so this reduces to a equals two."},{"Start":"03:41.750 ","End":"03:43.595","Text":"We found a and b,"},{"Start":"03:43.595 ","End":"03:46.700","Text":"and that solves this example,"},{"Start":"03:46.700 ","End":"03:49.080","Text":"and that\u0027s the end of this clip."}],"ID":28754},{"Watched":false,"Name":"Exercise 1 - Part a","Duration":"6m 19s","ChapterTopicVideoID":27551,"CourseChapterTopicPlaylistID":294457,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.690","Text":"In this exercise, we define a function g(x) on the interval from minus Pi to Pi,"},{"Start":"00:06.690 ","End":"00:08.535","Text":"like in the sketch."},{"Start":"00:08.535 ","End":"00:14.125","Text":"In part a, we have to compute the real Fourier series for g. Part b,"},{"Start":"00:14.125 ","End":"00:16.329","Text":"we define another function,"},{"Start":"00:16.329 ","End":"00:19.662","Text":"h(x) by the following formula,"},{"Start":"00:19.662 ","End":"00:21.645","Text":"where the g here,"},{"Start":"00:21.645 ","End":"00:25.350","Text":"is the same as the g here,"},{"Start":"00:25.350 ","End":"00:30.000","Text":"and there\u0027s a parameter a and the question is,"},{"Start":"00:30.000 ","End":"00:33.000","Text":"for which values of a does"},{"Start":"00:33.000 ","End":"00:38.220","Text":"the Fourier series of h converge uniformly to h on the interval."},{"Start":"00:38.220 ","End":"00:40.760","Text":"G(x) is of the following form,"},{"Start":"00:40.760 ","End":"00:44.425","Text":"which is the general form for a real Fourier series."},{"Start":"00:44.425 ","End":"00:48.240","Text":"Formula for a naught is this."},{"Start":"00:48.240 ","End":"00:52.790","Text":"What we\u0027ll do, is we\u0027ll break up the integral from minus Pi to Pi to 2 parts,"},{"Start":"00:52.790 ","End":"00:57.405","Text":"because it\u0027s defined separately from minus Pi to 0 and from 0 to Pi."},{"Start":"00:57.405 ","End":"00:58.770","Text":"Here it\u0027s minus x,"},{"Start":"00:58.770 ","End":"01:00.990","Text":"here it\u0027s Pi minus x."},{"Start":"01:00.990 ","End":"01:06.320","Text":"The minus x part appears both here and here."},{"Start":"01:06.320 ","End":"01:09.910","Text":"This part appears from minus Pi all the way up to Pi,"},{"Start":"01:09.910 ","End":"01:12.260","Text":"so we can take this part separately and get the"},{"Start":"01:12.260 ","End":"01:15.155","Text":"integral from minus Pi to Pi of minus x dx."},{"Start":"01:15.155 ","End":"01:17.165","Text":"All that\u0027s left, is the Pi here,"},{"Start":"01:17.165 ","End":"01:19.120","Text":"from 0 to Pi."},{"Start":"01:19.120 ","End":"01:23.795","Text":"This integral is 0, an odd function on a symmetric interval."},{"Start":"01:23.795 ","End":"01:25.160","Text":"This is a constant,"},{"Start":"01:25.160 ","End":"01:27.830","Text":"so the integral is Pi times x,"},{"Start":"01:27.830 ","End":"01:30.910","Text":"and between these 2 limits, is just Pi^2."},{"Start":"01:30.910 ","End":"01:34.395","Text":"Divide it by Pi, so the answer is Pi."},{"Start":"01:34.395 ","End":"01:36.360","Text":"That\u0027s a naught."},{"Start":"01:36.360 ","End":"01:39.405","Text":"Now we have to do a_n and b_n."},{"Start":"01:39.405 ","End":"01:42.600","Text":"This is the formula for a_n."},{"Start":"01:42.600 ","End":"01:46.170","Text":"Again we\u0027ll break up the integral into 2 integrals,"},{"Start":"01:46.170 ","End":"01:47.730","Text":"from minus Pi to 0,"},{"Start":"01:47.730 ","End":"01:48.990","Text":"and from 0 to Pi."},{"Start":"01:48.990 ","End":"01:50.100","Text":"Here it\u0027s minus x,"},{"Start":"01:50.100 ","End":"01:51.765","Text":"here it\u0027s Pi minus x."},{"Start":"01:51.765 ","End":"01:54.860","Text":"Just like before, we\u0027ll combine to the minus x,"},{"Start":"01:54.860 ","End":"01:56.090","Text":"both here and here,"},{"Start":"01:56.090 ","End":"02:00.020","Text":"so that minus x goes all the way from minus Pi to Pi."},{"Start":"02:00.020 ","End":"02:02.465","Text":"That\u0027s this first part."},{"Start":"02:02.465 ","End":"02:06.115","Text":"The second part we have just the Pi from 0 to Pi."},{"Start":"02:06.115 ","End":"02:08.580","Text":"Now this is 0,"},{"Start":"02:08.580 ","End":"02:10.920","Text":"because minus x is an odd function,"},{"Start":"02:10.920 ","End":"02:12.030","Text":"times an even function,"},{"Start":"02:12.030 ","End":"02:13.180","Text":"that\u0027s still an odd function,"},{"Start":"02:13.180 ","End":"02:15.595","Text":"on a symmetric interval that\u0027s 0."},{"Start":"02:15.595 ","End":"02:19.065","Text":"The other integral is this,"},{"Start":"02:19.065 ","End":"02:22.845","Text":"the Pi cancels with the 1 over Pi,"},{"Start":"02:22.845 ","End":"02:26.050","Text":"so we have the integral of cosine nx(dx)."},{"Start":"02:26.050 ","End":"02:32.330","Text":"That is equal to integral the sine nx over n. When you plug in Pi or 0,"},{"Start":"02:32.330 ","End":"02:36.125","Text":"we get 0, so the answer is 0 for this."},{"Start":"02:36.125 ","End":"02:40.615","Text":"We found a_n which equals 0."},{"Start":"02:40.615 ","End":"02:42.660","Text":"Next is b_n,"},{"Start":"02:42.660 ","End":"02:45.210","Text":"which is given by this formula,"},{"Start":"02:45.210 ","End":"02:49.860","Text":"and that is equal to as before we break this integral,"},{"Start":"02:49.860 ","End":"02:52.230","Text":"up into 2 from minus Pi to 0,"},{"Start":"02:52.230 ","End":"02:53.745","Text":"and from 0 to Pi."},{"Start":"02:53.745 ","End":"02:57.790","Text":"We get the integral of minus x all the way from minus Pi to Pi,"},{"Start":"02:57.790 ","End":"02:59.715","Text":"and then integral of Pi,"},{"Start":"02:59.715 ","End":"03:02.385","Text":"sine nx just from 0 to Pi."},{"Start":"03:02.385 ","End":"03:04.440","Text":"We\u0027ll do this by parts,"},{"Start":"03:04.440 ","End":"03:07.550","Text":"here is a formula for integration by parts."},{"Start":"03:07.550 ","End":"03:11.450","Text":"Integral of uv\u0027 is uv minus integral of u\u0027v."},{"Start":"03:11.450 ","End":"03:12.770","Text":"Sometimes it\u0027s the other way around."},{"Start":"03:12.770 ","End":"03:15.200","Text":"Here\u0027s the prime, here\u0027s naught, doesn\u0027t matter."},{"Start":"03:15.200 ","End":"03:21.065","Text":"We\u0027ll let u(x) be x and v\u0027(x) be minus sine(x)."},{"Start":"03:21.065 ","End":"03:23.420","Text":"It\u0027s convenient to take the minus here because the"},{"Start":"03:23.420 ","End":"03:26.405","Text":"integral of minus sine comes out to be cosine."},{"Start":"03:26.405 ","End":"03:28.610","Text":"Cosine nx over n,"},{"Start":"03:28.610 ","End":"03:31.025","Text":"that\u0027s v, this is u,"},{"Start":"03:31.025 ","End":"03:36.030","Text":"here\u0027s u\u0027, and here\u0027s v. That\u0027s for the first bit."},{"Start":"03:36.030 ","End":"03:37.950","Text":"Then there is second integral,"},{"Start":"03:37.950 ","End":"03:40.590","Text":"again, the Pi with the Pi cancels."},{"Start":"03:40.590 ","End":"03:44.410","Text":"It\u0027s just the integral of sine nx dx."},{"Start":"03:45.050 ","End":"03:48.025","Text":"Here if we substitute Pi,"},{"Start":"03:48.025 ","End":"03:50.960","Text":"we get this substitute minus Pi,"},{"Start":"03:50.960 ","End":"03:58.450","Text":"we get this, this integral is sine nx over n^2."},{"Start":"03:58.450 ","End":"04:04.140","Text":"This integral is minus cosine nx over n. Now cosine (n) Pi,"},{"Start":"04:04.140 ","End":"04:06.540","Text":"we know is minus 1^n."},{"Start":"04:06.540 ","End":"04:11.540","Text":"Similarly, cosine of minus n Pi is minus 1 to the minus n,"},{"Start":"04:11.540 ","End":"04:13.790","Text":"which is the same as minus 1^n."},{"Start":"04:13.790 ","End":"04:21.215","Text":"This is 0 because sine(n) Pi and sine of minus n Pi both 0. What do we get?"},{"Start":"04:21.215 ","End":"04:26.815","Text":"From this and this together we have twice because there\u0027s 2 of these that are the same."},{"Start":"04:26.815 ","End":"04:30.995","Text":"Each of these is Pi cosine n Pi over n. In other words,"},{"Start":"04:30.995 ","End":"04:35.120","Text":"Pi minus 1 to the n over n. That\u0027s these 2 pieces."},{"Start":"04:35.120 ","End":"04:38.825","Text":"This part is 0 and we\u0027re left with this part."},{"Start":"04:38.825 ","End":"04:43.310","Text":"Instead of the minus will take the 0 on top and the Pi on the bottom,"},{"Start":"04:43.310 ","End":"04:46.130","Text":"so cosine 0 is 1."},{"Start":"04:46.130 ","End":"04:51.150","Text":"Cosine n Pi, again is minus 1 to the n over n. Now,"},{"Start":"04:51.150 ","End":"04:54.675","Text":"this Pi with this Pi cancels."},{"Start":"04:54.675 ","End":"05:02.240","Text":"Here we have twice minus 1 to the n over n. Here we have minus the same thing."},{"Start":"05:02.240 ","End":"05:04.120","Text":"We have 2 of these minus 1 of these."},{"Start":"05:04.120 ","End":"05:08.920","Text":"It just gives us 1 of these that makes this a plus here instead of a minus,"},{"Start":"05:08.920 ","End":"05:11.160","Text":"and this will divide up into 2 cases."},{"Start":"05:11.160 ","End":"05:13.055","Text":"N is odd or n is even."},{"Start":"05:13.055 ","End":"05:18.235","Text":"If n is even minus 1 to the n becomes 1."},{"Start":"05:18.235 ","End":"05:21.185","Text":"We have 2 over n, but n is 2k."},{"Start":"05:21.185 ","End":"05:23.810","Text":"That\u0027s the general even number."},{"Start":"05:23.810 ","End":"05:25.535","Text":"If n is odd,"},{"Start":"05:25.535 ","End":"05:28.085","Text":"we have 1 minus 1 over n is 0."},{"Start":"05:28.085 ","End":"05:30.070","Text":"This is what we have now."},{"Start":"05:30.070 ","End":"05:32.130","Text":"The 2 with the 2 cancels,"},{"Start":"05:32.130 ","End":"05:35.460","Text":"this is just 1 over k. Summarizing,"},{"Start":"05:35.460 ","End":"05:38.220","Text":"a naught is Pi a_n,"},{"Start":"05:38.220 ","End":"05:40.665","Text":"where n is not 0, is 0,"},{"Start":"05:40.665 ","End":"05:48.720","Text":"b_n from here is 1 over k if n is 2k and 0 if n is 2k minus 1."},{"Start":"05:48.720 ","End":"05:54.860","Text":"Now substitute in the general Fourier series, we have everything."},{"Start":"05:54.860 ","End":"05:57.235","Text":"We have a naught a_n and b_n."},{"Start":"05:57.235 ","End":"06:00.345","Text":"This part is all zeros."},{"Start":"06:00.345 ","End":"06:01.860","Text":"We have a naught over 2,"},{"Start":"06:01.860 ","End":"06:03.600","Text":"which is Pi over 2."},{"Start":"06:03.600 ","End":"06:07.890","Text":"Here, b_n is 1 over k,"},{"Start":"06:07.890 ","End":"06:09.480","Text":"just 1 over k sine(n),"},{"Start":"06:09.480 ","End":"06:11.535","Text":"but n is 2k."},{"Start":"06:11.535 ","End":"06:13.880","Text":"Also k goes from 1 to infinity."},{"Start":"06:13.880 ","End":"06:16.789","Text":"That covers all the even numbers."},{"Start":"06:16.789 ","End":"06:19.950","Text":"This is the answer for part a."}],"ID":28755},{"Watched":false,"Name":"Exercise 1 - Part b","Duration":"3m 10s","ChapterTopicVideoID":27552,"CourseChapterTopicPlaylistID":294457,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"Continuing with this exercise, Part B,"},{"Start":"00:03.120 ","End":"00:04.785","Text":"we define another function,"},{"Start":"00:04.785 ","End":"00:08.745","Text":"h of x by the following formula,"},{"Start":"00:08.745 ","End":"00:15.210","Text":"where the g here is the same as the g in here,"},{"Start":"00:15.210 ","End":"00:20.730","Text":"part A and there\u0027s a parameter a."},{"Start":"00:20.730 ","End":"00:25.670","Text":"The question is, for which values of a does"},{"Start":"00:25.670 ","End":"00:31.265","Text":"the Fourier series of h converge uniformly to h on the interval?"},{"Start":"00:31.265 ","End":"00:34.790","Text":"Recall the following theorem or proposition,"},{"Start":"00:34.790 ","End":"00:37.370","Text":"for given function f such that f and its"},{"Start":"00:37.370 ","End":"00:42.200","Text":"derivative are both piecewise continuous on minus Pi,"},{"Start":"00:42.200 ","End":"00:48.080","Text":"Pi and f is continuous and f of minus Pi equals f of Pi,"},{"Start":"00:48.080 ","End":"00:53.750","Text":"then the Fourier series for f converges uniformly to f on the interval."},{"Start":"00:53.750 ","End":"01:02.775","Text":"We want to apply this theorem to our function h. Let\u0027s find a such that this is true."},{"Start":"01:02.775 ","End":"01:06.015","Text":"H of minus Pi equals h of Pi."},{"Start":"01:06.015 ","End":"01:10.020","Text":"H of Pi is given by this,"},{"Start":"01:10.020 ","End":"01:13.280","Text":"just what we get if we substitute x equals Pi here,"},{"Start":"01:13.280 ","End":"01:15.140","Text":"we substituted here and here,"},{"Start":"01:15.140 ","End":"01:17.450","Text":"and this integral, well,"},{"Start":"01:17.450 ","End":"01:19.375","Text":"you can do it with a picture,"},{"Start":"01:19.375 ","End":"01:25.075","Text":"it\u0027s the area under the curve g from minus Pi to Pi."},{"Start":"01:25.075 ","End":"01:28.550","Text":"Basically, it\u0027s the same as the area of this square."},{"Start":"01:28.550 ","End":"01:29.930","Text":"Put the two triangles together,"},{"Start":"01:29.930 ","End":"01:31.310","Text":"we have a square,"},{"Start":"01:31.310 ","End":"01:33.875","Text":"width Pi, height Pi,"},{"Start":"01:33.875 ","End":"01:35.905","Text":"so it\u0027s Pi squared."},{"Start":"01:35.905 ","End":"01:37.335","Text":"H of minus Pi,"},{"Start":"01:37.335 ","End":"01:40.370","Text":"the same thing except we have a minus Pi here and here,"},{"Start":"01:40.370 ","End":"01:43.279","Text":"and this integral is 0."},{"Start":"01:43.279 ","End":"01:47.285","Text":"The integral from something to itself is always 0 and"},{"Start":"01:47.285 ","End":"01:52.490","Text":"sine of minus Pi over 2 is minus 1 so we get minus a."},{"Start":"01:52.490 ","End":"01:55.780","Text":"Now we have to compare these two."},{"Start":"01:55.780 ","End":"02:00.995","Text":"These two are equal if and only if a plus Pi squared is the same as minus a,"},{"Start":"02:00.995 ","End":"02:06.410","Text":"which gives us that 2a is minus Pi squared or a is minus Pi squared over 2."},{"Start":"02:06.410 ","End":"02:08.750","Text":"This choice will guarantee that this equals this."},{"Start":"02:08.750 ","End":"02:11.015","Text":"Let\u0027s make sure the other conditions hold."},{"Start":"02:11.015 ","End":"02:12.950","Text":"H of x is now equal to,"},{"Start":"02:12.950 ","End":"02:14.180","Text":"put instead of a,"},{"Start":"02:14.180 ","End":"02:16.570","Text":"we put minus Pi squared over 2."},{"Start":"02:16.570 ","End":"02:18.975","Text":"We already checked this."},{"Start":"02:18.975 ","End":"02:23.780","Text":"Other condition is that it\u0027s continuous and that\u0027s true because it\u0027s differentiable."},{"Start":"02:23.780 ","End":"02:25.820","Text":"This part is differentiable,"},{"Start":"02:25.820 ","End":"02:27.590","Text":"and this part is differentiable,"},{"Start":"02:27.590 ","End":"02:33.020","Text":"it\u0027s derivative is g. The last condition is that h\u0027"},{"Start":"02:33.020 ","End":"02:38.480","Text":"is piecewise continuous and that\u0027s true because h\u0027 is,"},{"Start":"02:38.480 ","End":"02:43.010","Text":"differentiate this, we get cosine of x over 2 and then times"},{"Start":"02:43.010 ","End":"02:48.710","Text":"a 1/2 plus derivative of this is g of x by the fundamental theorem of calculus."},{"Start":"02:48.710 ","End":"02:52.150","Text":"The derivative of the integral is the function itself."},{"Start":"02:52.150 ","End":"02:54.350","Text":"g is piecewise continuous."},{"Start":"02:54.350 ","End":"02:55.880","Text":"This is continuous altogether,"},{"Start":"02:55.880 ","End":"02:58.475","Text":"so yeah so h\u0027 is piecewise continuous."},{"Start":"02:58.475 ","End":"03:02.570","Text":"We\u0027ve met all the conditions and so the Fourier series for"},{"Start":"03:02.570 ","End":"03:07.715","Text":"h converges to h because h satisfies the conditions of the theorem."},{"Start":"03:07.715 ","End":"03:10.740","Text":"That concludes this exercise."}],"ID":28756},{"Watched":false,"Name":"Exercise 2 - Part a","Duration":"6m 13s","ChapterTopicVideoID":27553,"CourseChapterTopicPlaylistID":294457,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.595","Text":"In this exercise, we consider the function f(x) equals absolute value of sine x."},{"Start":"00:05.595 ","End":"00:07.320","Text":"This is the picture,"},{"Start":"00:07.320 ","End":"00:11.760","Text":"in the space of piecewise continuous functions on minus pi,"},{"Start":"00:11.760 ","End":"00:14.865","Text":"pi and that is derivative would be f’(x)."},{"Start":"00:14.865 ","End":"00:17.790","Text":"This is also piecewise continuous."},{"Start":"00:17.790 ","End":"00:20.090","Text":"I mean, it doesn\u0027t have a derivative,"},{"Start":"00:20.090 ","End":"00:21.960","Text":"f\u0027 doesn\u0027t exist at this point,"},{"Start":"00:21.960 ","End":"00:24.195","Text":"but piecewise it does."},{"Start":"00:24.195 ","End":"00:27.930","Text":"In Part A, we have to compute the real Fourier series for f"},{"Start":"00:27.930 ","End":"00:32.250","Text":"and for f\u0027 and read the other parts has become to them."},{"Start":"00:32.250 ","End":"00:37.830","Text":"The general form of a real Fourier series for a function on minus pi,"},{"Start":"00:37.830 ","End":"00:39.720","Text":"pi is the following,"},{"Start":"00:39.720 ","End":"00:42.360","Text":"but our function is even."},{"Start":"00:42.360 ","End":"00:46.395","Text":"We could straightaway throughout the b_n sine(nx),"},{"Start":"00:46.395 ","End":"00:50.645","Text":"we just have a_ns, and we\u0027ll use the standard formulas for these."},{"Start":"00:50.645 ","End":"00:54.530","Text":"The formula for a_n is this and let\u0027s compute it."},{"Start":"00:54.530 ","End":"00:56.875","Text":"Because it\u0027s an even function,"},{"Start":"00:56.875 ","End":"01:03.710","Text":"even times even is even so we can double the integral and just take it from 0 to pi."},{"Start":"01:03.710 ","End":"01:08.060","Text":"f’(x) is equal to sine x on the interval from 0 to pi."},{"Start":"01:08.060 ","End":"01:10.870","Text":"Don\u0027t need the absolute value for this part."},{"Start":"01:10.870 ","End":"01:13.655","Text":"We\u0027ll use this trigonometric identity."},{"Start":"01:13.655 ","End":"01:17.116","Text":"Beta is x and Alpha is nx,"},{"Start":"01:17.116 ","End":"01:21.180","Text":"and using this formula, we get this."},{"Start":"01:21.180 ","End":"01:23.450","Text":"I\u0027ll leave you to check the details."},{"Start":"01:23.450 ","End":"01:27.920","Text":"Now the integral of sine is minus cosine."},{"Start":"01:27.920 ","End":"01:33.350","Text":"What we\u0027re going to do to save a step is not write it as minus cosine but just as"},{"Start":"01:33.350 ","End":"01:38.420","Text":"cosine and compensate for the minus by reversing the limits of integration,"},{"Start":"01:38.420 ","End":"01:40.250","Text":"take it from pi to 0."},{"Start":"01:40.250 ","End":"01:44.750","Text":"Cosine of n plus 1 is x over n plus 1 from pi to 0."},{"Start":"01:44.750 ","End":"01:49.950","Text":"When x is 0, we just get 1 and when x is pi,"},{"Start":"01:49.950 ","End":"01:52.980","Text":"we get cosine n plus 1 pi,"},{"Start":"01:52.980 ","End":"01:55.730","Text":"which is minus 1 to the n plus 1,"},{"Start":"01:55.730 ","End":"01:59.600","Text":"but it has a minus in front of it because it\u0027s the lower limit."},{"Start":"01:59.600 ","End":"02:01.370","Text":"So we have 1 minus,"},{"Start":"02:01.370 ","End":"02:08.110","Text":"minus 1 to the n plus 1 and that makes it the same as 1 plus minus 1^n."},{"Start":"02:08.110 ","End":"02:10.130","Text":"Similarly with the other one,"},{"Start":"02:10.130 ","End":"02:13.360","Text":"we have minus 1^n minus 1,"},{"Start":"02:13.360 ","End":"02:19.445","Text":"but it\u0027s 1 minus that so we can plus and then change the index by 1."},{"Start":"02:19.445 ","End":"02:22.970","Text":"In both cases we have minus 1^n."},{"Start":"02:22.970 ","End":"02:26.891","Text":"That gives us 1 plus minus 1^n,"},{"Start":"02:26.891 ","End":"02:28.550","Text":"that is 1 over pi,"},{"Start":"02:28.550 ","End":"02:32.150","Text":"and then this 1 over n plus 1 minus 1 over n minus 1."},{"Start":"02:32.150 ","End":"02:33.590","Text":"This we can subtract it,"},{"Start":"02:33.590 ","End":"02:37.710","Text":"cross-multiply and we have n minus 1,"},{"Start":"02:37.710 ","End":"02:42.410","Text":"subtract n plus 1 gives us minus 2 over the product."},{"Start":"02:42.410 ","End":"02:44.975","Text":"Put the pi on the denominator,"},{"Start":"02:44.975 ","End":"02:46.835","Text":"this is what we have now."},{"Start":"02:46.835 ","End":"02:50.855","Text":"We\u0027ll separate cases: n even and n odd."},{"Start":"02:50.855 ","End":"02:56.498","Text":"I forgot to say earlier that here we have to say that n is not equal to 1."},{"Start":"02:56.498 ","End":"02:59.809","Text":"Separate into odds and evens."},{"Start":"02:59.809 ","End":"03:05.660","Text":"For even, we get minus 2 from here and 1 plus 1 is 2 from here,"},{"Start":"03:05.660 ","End":"03:11.500","Text":"so it\u0027s minus 4 over pi and replace n by 2k here."},{"Start":"03:11.500 ","End":"03:20.110","Text":"For n odd, it\u0027s 0 because 1 plus minus 1 is 0. That\u0027s a_n."},{"Start":"03:20.110 ","End":"03:22.760","Text":"Next, let\u0027s compute what a_1 is,"},{"Start":"03:22.760 ","End":"03:25.355","Text":"because that\u0027s an exception case."},{"Start":"03:25.355 ","End":"03:30.995","Text":"Let\u0027s go back to the formula for a_n and replace n by 1."},{"Start":"03:30.995 ","End":"03:33.640","Text":"This is the integral we have."},{"Start":"03:33.640 ","End":"03:37.020","Text":"Sine x, cosine x."},{"Start":"03:37.020 ","End":"03:40.070","Text":"Here we can use a trigonometric formula,"},{"Start":"03:40.070 ","End":"03:43.700","Text":"2 sine x cosine x is sine 2x."},{"Start":"03:43.700 ","End":"03:45.755","Text":"The part colored in green,"},{"Start":"03:45.755 ","End":"03:48.740","Text":"it\u0027s a standard trigonometric identity."},{"Start":"03:48.740 ","End":"03:53.345","Text":"To do this integral will make a substitution that t equal to x"},{"Start":"03:53.345 ","End":"04:00.735","Text":"so that dt is 2dx and x goes from 0 to pi,"},{"Start":"04:00.735 ","End":"04:06.095","Text":"so t goes from 0 to 2pi because t is twice x."},{"Start":"04:06.095 ","End":"04:08.330","Text":"What we get after the substitution is"},{"Start":"04:08.330 ","End":"04:13.760","Text":"this integral over 1 whole period of sine, and we know that\u0027s 0."},{"Start":"04:13.760 ","End":"04:15.720","Text":"It\u0027s easy enough to compute."},{"Start":"04:15.720 ","End":"04:21.110","Text":"What this means is that the formula for a_n stands and we don\u0027t have"},{"Start":"04:21.110 ","End":"04:27.010","Text":"to exclude n equals 1 because n equals 1 is included in the odd case and it is 0."},{"Start":"04:27.010 ","End":"04:29.645","Text":"That works out nicely."},{"Start":"04:29.645 ","End":"04:32.360","Text":"That\u0027s what a_n is for n equals 0,"},{"Start":"04:32.360 ","End":"04:35.405","Text":"1, 2, 3, etc and the bn\u0027s are all 0."},{"Start":"04:35.405 ","End":"04:38.360","Text":"Now a_0 we treat a bit differently."},{"Start":"04:38.360 ","End":"04:45.990","Text":"A_0 is minus 4 over pi times 1 times minus 1,"},{"Start":"04:45.990 ","End":"04:48.270","Text":"so it comes out 4 over pi."},{"Start":"04:48.270 ","End":"04:52.125","Text":"Because in the formula a_0 has to be divided by 2."},{"Start":"04:52.125 ","End":"04:53.880","Text":"Instead of 4 over pi,"},{"Start":"04:53.880 ","End":"04:58.610","Text":"it\u0027s just 2 over pi and the rest of it we just put a_n,"},{"Start":"04:58.610 ","End":"05:00.350","Text":"as it was here."},{"Start":"05:00.350 ","End":"05:02.600","Text":"Just as a by the way,"},{"Start":"05:02.600 ","End":"05:04.370","Text":"the a naught over 2,"},{"Start":"05:04.370 ","End":"05:06.080","Text":"the constant term here,"},{"Start":"05:06.080 ","End":"05:10.220","Text":"in the Fourier series is the average of the function over the interval."},{"Start":"05:10.220 ","End":"05:13.460","Text":"In this case, 2 over pi is the average of"},{"Start":"05:13.460 ","End":"05:17.480","Text":"the function I thought I\u0027d mentioned for interest\u0027s sake."},{"Start":"05:17.480 ","End":"05:21.710","Text":"That\u0027s the Fourier series for f. We still have to do"},{"Start":"05:21.710 ","End":"05:26.860","Text":"f\u0027 and we\u0027ll do this by term by term differentiation."},{"Start":"05:26.860 ","End":"05:30.965","Text":"Now the conditions for differentiating term by term are met."},{"Start":"05:30.965 ","End":"05:35.045","Text":"FA is continuous, f of minus pi is f of pi,"},{"Start":"05:35.045 ","End":"05:38.095","Text":"and f\u0027 is piecewise continuous."},{"Start":"05:38.095 ","End":"05:41.870","Text":"F\u0027 isn\u0027t continuous, it isn\u0027t even defined at 0"},{"Start":"05:41.870 ","End":"05:46.190","Text":"once you do f\u0027 on these open interval from minus pi to 0,"},{"Start":"05:46.190 ","End":"05:49.265","Text":"from 0 to pi, and it is continuous."},{"Start":"05:49.265 ","End":"05:51.675","Text":"We have all the conditions here."},{"Start":"05:51.675 ","End":"05:57.210","Text":"Term wise differentiation gives us minus 4 times"},{"Start":"05:57.210 ","End":"06:02.690","Text":"2_k times minus 1 because derivative of cosine is minus sine,"},{"Start":"06:02.690 ","End":"06:07.970","Text":"we get 8_k and the denominator is the same,"},{"Start":"06:07.970 ","End":"06:11.672","Text":"and this is f’(x)."},{"Start":"06:11.672 ","End":"06:14.920","Text":"So that concludes this exercise."}],"ID":28757},{"Watched":false,"Name":"Exercise 2 - Part b","Duration":"2m 55s","ChapterTopicVideoID":27554,"CourseChapterTopicPlaylistID":294457,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:02.940","Text":"Continuing, with just done part A,"},{"Start":"00:02.940 ","End":"00:04.500","Text":"now part B,"},{"Start":"00:04.500 ","End":"00:10.440","Text":"to which functions do the series for f and f prime converge."},{"Start":"00:10.440 ","End":"00:12.330","Text":"We have to sketch the graphs."},{"Start":"00:12.330 ","End":"00:17.599","Text":"In part A we found that the Fourier series for f is this."},{"Start":"00:17.599 ","End":"00:21.770","Text":"Now the question of convergence f is continuous,"},{"Start":"00:21.770 ","End":"00:24.590","Text":"so by Dirichlet\u0027s theorem,"},{"Start":"00:24.590 ","End":"00:28.190","Text":"f(x) is equal to the series."},{"Start":"00:28.190 ","End":"00:31.475","Text":"It actually holds for the periodic extension,"},{"Start":"00:31.475 ","End":"00:36.675","Text":"call it f tilde of f on all of R. The formula stays the same,"},{"Start":"00:36.675 ","End":"00:43.085","Text":"it\u0027s still absolute value of sine x and here\u0027s the picture of it."},{"Start":"00:43.085 ","End":"00:47.795","Text":"This is the original f from minus Pi to Pi,"},{"Start":"00:47.795 ","End":"00:50.090","Text":"and you can extend it indefinitely."},{"Start":"00:50.090 ","End":"00:53.450","Text":"But the sketch we just had to do from minus 3 Pi to 3 Pi."},{"Start":"00:53.450 ","End":"00:57.895","Text":"It\u0027s basically a copy paste of this here and here."},{"Start":"00:57.895 ","End":"01:01.505","Text":"That was f, now what about f prime?"},{"Start":"01:01.505 ","End":"01:05.064","Text":"F prime is equal to this"},{"Start":"01:05.064 ","End":"01:09.650","Text":"on the interval from minus Pi to Pi and this is what it looks like."},{"Start":"01:09.650 ","End":"01:11.360","Text":"This is the cosine,"},{"Start":"01:11.360 ","End":"01:16.430","Text":"but we take it negative from minus Pi to 0,"},{"Start":"01:16.430 ","End":"01:19.130","Text":"just the mirror image of this."},{"Start":"01:19.130 ","End":"01:22.205","Text":"It\u0027s not continuous at 0,"},{"Start":"01:22.205 ","End":"01:25.805","Text":"not continuous multiples of Pi."},{"Start":"01:25.805 ","End":"01:28.460","Text":"At these points of discontinuity,"},{"Start":"01:28.460 ","End":"01:32.930","Text":"we have a limit on the right and the left and theorem says"},{"Start":"01:32.930 ","End":"01:37.550","Text":"that it converges to the average this plus this over 2 here."},{"Start":"01:37.550 ","End":"01:39.855","Text":"We have Pi here minus Pi, so this is 0."},{"Start":"01:39.855 ","End":"01:43.160","Text":"Similarly here and here,"},{"Start":"01:43.160 ","End":"01:45.395","Text":"the series converges to 0."},{"Start":"01:45.395 ","End":"01:48.440","Text":"Now this series converges to a function,"},{"Start":"01:48.440 ","End":"01:51.785","Text":"let\u0027s call it f prime with a tilde over it,"},{"Start":"01:51.785 ","End":"01:57.755","Text":"where this is equal to minus cosine x and not just from minus Pi to 0,"},{"Start":"01:57.755 ","End":"02:01.670","Text":"we can add 2 Pi or multiples of 2 Pi to this."},{"Start":"02:01.670 ","End":"02:06.515","Text":"In general, it\u0027ll be 2k minus 1 Pi, 2kPi."},{"Start":"02:06.515 ","End":"02:10.760","Text":"Similarly here, is what we have here,"},{"Start":"02:10.760 ","End":"02:14.360","Text":"but we add multiples of 2 Pi."},{"Start":"02:14.360 ","End":"02:18.145","Text":"Also at multiples of Pi,"},{"Start":"02:18.145 ","End":"02:22.260","Text":"you could say it\u0027s 2kPi and 2k plus 1 Pi,"},{"Start":"02:22.260 ","End":"02:27.230","Text":"but these combined, this is even number times Pi or odd number times Pi."},{"Start":"02:27.230 ","End":"02:30.620","Text":"In general, it\u0027s just an integer times pi."},{"Start":"02:30.620 ","End":"02:33.184","Text":"There, it\u0027s 0. Let\u0027s see the picture."},{"Start":"02:33.184 ","End":"02:36.525","Text":"Here, it\u0027s cosine,"},{"Start":"02:36.525 ","End":"02:38.835","Text":"and also here It\u0027s cosine,"},{"Start":"02:38.835 ","End":"02:40.635","Text":"and here it\u0027s cosine,"},{"Start":"02:40.635 ","End":"02:43.115","Text":"and then here it\u0027s minus cosine,"},{"Start":"02:43.115 ","End":"02:45.085","Text":"here it\u0027s minus cosine,"},{"Start":"02:45.085 ","End":"02:47.585","Text":"and here it\u0027s minus cosine."},{"Start":"02:47.585 ","End":"02:52.655","Text":"Also at these points multiples of Pi, it\u0027s exactly 0."},{"Start":"02:52.655 ","End":"02:56.010","Text":"That concludes part B."}],"ID":28758},{"Watched":false,"Name":"Exercise 2 - Part c","Duration":"2m 23s","ChapterTopicVideoID":27549,"CourseChapterTopicPlaylistID":294457,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"Now we come to part C of the exercise."},{"Start":"00:03.270 ","End":"00:06.030","Text":"We have to say on which closed intervals do"},{"Start":"00:06.030 ","End":"00:10.950","Text":"the series for f and f prime converge uniformly?"},{"Start":"00:10.950 ","End":"00:15.990","Text":"We saw in part B that the series converges to f tilde,"},{"Start":"00:15.990 ","End":"00:19.005","Text":"which is the extension of f to all the real line."},{"Start":"00:19.005 ","End":"00:22.245","Text":"This is just regular pointwise convergence."},{"Start":"00:22.245 ","End":"00:26.115","Text":"Now, this f tilde is continuous,"},{"Start":"00:26.115 ","End":"00:33.230","Text":"2Pi periodic and its derivative is piecewise continuous on all R. I mean,"},{"Start":"00:33.230 ","End":"00:36.970","Text":"if you remove the points nPi multiples of Pi,"},{"Start":"00:36.970 ","End":"00:39.090","Text":"then it is continuous,"},{"Start":"00:39.090 ","End":"00:41.315","Text":"but as it is, it\u0027s piecewise."},{"Start":"00:41.315 ","End":"00:48.740","Text":"The Fourier series of f converges to it uniformly on all of R in whole real line."},{"Start":"00:48.740 ","End":"00:53.780","Text":"That takes care of the function f. Now what about f prime?"},{"Start":"00:53.780 ","End":"00:55.490","Text":"Well, we had the series for"},{"Start":"00:55.490 ","End":"01:02.195","Text":"f prime periodic extension was this and this is what it looks like."},{"Start":"01:02.195 ","End":"01:07.235","Text":"We see that, except that multiples of Pi,"},{"Start":"01:07.235 ","End":"01:09.740","Text":"this is not only continuous,"},{"Start":"01:09.740 ","End":"01:12.355","Text":"but it also has a continuous derivative,"},{"Start":"01:12.355 ","End":"01:14.990","Text":"so f prime minus derivative of piecewise"},{"Start":"01:14.990 ","End":"01:19.400","Text":"continuous and its extension is continuous except at"},{"Start":"01:19.400 ","End":"01:21.740","Text":"multiples of Pi and there\u0027s"},{"Start":"01:21.740 ","End":"01:26.405","Text":"a generalized proposition that says that under these conditions,"},{"Start":"01:26.405 ","End":"01:28.735","Text":"we have uniform convergence."},{"Start":"01:28.735 ","End":"01:36.350","Text":"So what we have to do is say which closed intervals exclude multiples of Pi?"},{"Start":"01:36.350 ","End":"01:39.785","Text":"Such an interval will have the following formula."},{"Start":"01:39.785 ","End":"01:42.485","Text":"Well, let\u0027s see a picture that will explain it."},{"Start":"01:42.485 ","End":"01:47.030","Text":"We go from nPi to n plus 1Pi,"},{"Start":"01:47.030 ","End":"01:52.565","Text":"but we have to exclude this so we take a bit off here and a bit off here,"},{"Start":"01:52.565 ","End":"01:57.610","Text":"let\u0027s take Delta off and then it will be given by this formula."},{"Start":"01:57.610 ","End":"02:01.790","Text":"Actually, this is not quite precise because we don\u0027t have to take the same Delta"},{"Start":"02:01.790 ","End":"02:05.874","Text":"here and here we could take Delta 1 here and Delta 2 here."},{"Start":"02:05.874 ","End":"02:11.270","Text":"Yes, this is more accurately phrased and basically that\u0027s it and it"},{"Start":"02:11.270 ","End":"02:17.255","Text":"closed the interval that\u0027s contained in this interval from nPi to n plus 1Pi,"},{"Start":"02:17.255 ","End":"02:20.920","Text":"it\u0027s closed and there it converges uniformly."},{"Start":"02:20.920 ","End":"02:24.020","Text":"That concludes this exercise."}],"ID":28759}],"Thumbnail":null,"ID":294457},{"Name":"Fourier Series on a General Interval","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Fourier Series on a General Interval + Example","Duration":"8m 41s","ChapterTopicVideoID":27579,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.685","Text":"Now it\u0027s time to generalize a bit."},{"Start":"00:02.685 ","End":"00:06.990","Text":"Up to now we\u0027ve been talking about the integral from minus Pi to Pi,"},{"Start":"00:06.990 ","End":"00:10.560","Text":"or almost the same from 0 to 2Pi."},{"Start":"00:10.560 ","End":"00:15.420","Text":"We want to generalize to any closed interval from a to b."},{"Start":"00:15.420 ","End":"00:18.540","Text":"We\u0027re not going to generalize to infinite intervals,"},{"Start":"00:18.540 ","End":"00:20.665","Text":"but just to finite ones."},{"Start":"00:20.665 ","End":"00:23.535","Text":"We\u0027re going to abbreviate the name of the space instead of"},{"Start":"00:23.535 ","End":"00:27.120","Text":"L^2_PC we\u0027ll just write it as E for convenience."},{"Start":"00:27.120 ","End":"00:30.900","Text":"What we\u0027ll do is go through the main formulas and theorems and see"},{"Start":"00:30.900 ","End":"00:35.070","Text":"how they adapt to a general interval a, b."},{"Start":"00:35.070 ","End":"00:40.580","Text":"In the product, we have 2 versions, real or complex."},{"Start":"00:40.580 ","End":"00:41.660","Text":"When I say real or complex,"},{"Start":"00:41.660 ","End":"00:44.510","Text":"not only in the function is necessarily a real or complex,"},{"Start":"00:44.510 ","End":"00:50.419","Text":"just that the orthonormal set is real or complex as the real version,"},{"Start":"00:50.419 ","End":"00:53.380","Text":"complex version of things."},{"Start":"00:53.380 ","End":"00:58.520","Text":"Formula is almost the same except that in the real case there\u0027s a 2 here."},{"Start":"00:58.520 ","End":"01:02.375","Text":"Note that if b minus a is 2Pi,"},{"Start":"01:02.375 ","End":"01:04.835","Text":"like it is here and here,"},{"Start":"01:04.835 ","End":"01:07.110","Text":"then here we have 1 over Pi,"},{"Start":"01:07.110 ","End":"01:09.370","Text":"and here we have 1 over 2Pi."},{"Start":"01:09.370 ","End":"01:11.810","Text":"Which is really the only difference between the real and"},{"Start":"01:11.810 ","End":"01:14.345","Text":"complex as far as in a product goes."},{"Start":"01:14.345 ","End":"01:17.140","Text":"Now, orthonormal system."},{"Start":"01:17.140 ","End":"01:21.200","Text":"The real orthonormal system is set of 1"},{"Start":"01:21.200 ","End":"01:25.495","Text":"over root 2 and then the cosine nx and sine nx set."},{"Start":"01:25.495 ","End":"01:29.670","Text":"That\u0027s for 0 to 2Pi or minus Pi to Pi."},{"Start":"01:29.670 ","End":"01:34.010","Text":"We adapt it by multiplying inside by a factor,"},{"Start":"01:34.010 ","End":"01:38.230","Text":"stretching 2Pi over b minus a."},{"Start":"01:38.230 ","End":"01:40.080","Text":"Instead of 2Pi interval,"},{"Start":"01:40.080 ","End":"01:42.110","Text":"we have b minus a length interval."},{"Start":"01:42.110 ","End":"01:48.325","Text":"So we stretch or compress by a factor of 2Pi over b minus a."},{"Start":"01:48.325 ","End":"01:51.125","Text":"Similarly, in the complex set,"},{"Start":"01:51.125 ","End":"01:56.470","Text":"everywhere we have this 2Pi over b minus a to scale it up or down."},{"Start":"01:56.470 ","End":"01:58.950","Text":"The Fourier series are similar."},{"Start":"01:58.950 ","End":"02:03.050","Text":"But again, we have the 2Pi over b minus a inside."},{"Start":"02:03.050 ","End":"02:05.285","Text":"Other than that, it looks pretty similar."},{"Start":"02:05.285 ","End":"02:09.530","Text":"We have a formula for the coefficients c_n in this case,"},{"Start":"02:09.530 ","End":"02:12.365","Text":"or a_n and b_n in this case,"},{"Start":"02:12.365 ","End":"02:15.020","Text":"in the complex case, this is c_n."},{"Start":"02:15.020 ","End":"02:18.230","Text":"In the real case,"},{"Start":"02:18.230 ","End":"02:21.845","Text":"we have separate formulas for a_0, a_n and b_n."},{"Start":"02:21.845 ","End":"02:28.240","Text":"Although I notice a special case of a_n with n equals 0 because cosine 0 is 1."},{"Start":"02:28.240 ","End":"02:33.980","Text":"Now let\u0027s get to the Dirichlet condition on pointwise convergence."},{"Start":"02:33.980 ","End":"02:36.740","Text":"The theorem or proposition is very similar to"},{"Start":"02:36.740 ","End":"02:41.600","Text":"the general case if x is in the open interval a, b,"},{"Start":"02:41.600 ","End":"02:47.720","Text":"and if it has finite left and right derivatives not necessarily equal to each other,"},{"Start":"02:47.720 ","End":"02:49.474","Text":"but both exist in finite."},{"Start":"02:49.474 ","End":"02:54.920","Text":"Then we have pointwise equality at x that the sum of"},{"Start":"02:54.920 ","End":"03:00.520","Text":"the series is the average between the right and left limit at the point."},{"Start":"03:00.520 ","End":"03:06.800","Text":"Similarly, for the real version of the Fourier complex real,"},{"Start":"03:06.800 ","End":"03:11.290","Text":"both cases equal to the average of the right and the left limits."},{"Start":"03:11.290 ","End":"03:14.290","Text":"Now uniform convergence."},{"Start":"03:14.290 ","End":"03:17.330","Text":"The theorem is practically the same."},{"Start":"03:17.330 ","End":"03:19.070","Text":"Instead of minus Pi to Pi,"},{"Start":"03:19.070 ","End":"03:20.270","Text":"we have a, b."},{"Start":"03:20.270 ","End":"03:22.850","Text":"If it\u0027s continuous and equal at"},{"Start":"03:22.850 ","End":"03:27.760","Text":"the endpoints and the derivative is piecewise continuous,"},{"Start":"03:27.760 ","End":"03:35.345","Text":"then the Fourier series for f converges uniformly to f on a, b. Parseval\u0027s identity."},{"Start":"03:35.345 ","End":"03:38.695","Text":"That\u0027s always true, no conditions."},{"Start":"03:38.695 ","End":"03:42.000","Text":"This is actually the norm squared."},{"Start":"03:42.000 ","End":"03:45.260","Text":"The inner product of f with itself is equal to"},{"Start":"03:45.260 ","End":"03:49.400","Text":"the sum of the absolute value of the coefficient squared."},{"Start":"03:49.400 ","End":"03:51.350","Text":"That\u0027s in the complex case."},{"Start":"03:51.350 ","End":"03:53.760","Text":"In the real case, again,"},{"Start":"03:53.760 ","End":"03:56.640","Text":"it\u0027s the norm squared only there is a 2 here,"},{"Start":"03:56.640 ","End":"04:01.115","Text":"and it equals to the sum of the coefficient squared in absolute value."},{"Start":"04:01.115 ","End":"04:07.100","Text":"The generalized Parseval\u0027s identity also works when we have 2 functions,"},{"Start":"04:07.100 ","End":"04:11.240","Text":"f and g. Instead of taking the inner product of f with itself,"},{"Start":"04:11.240 ","End":"04:16.270","Text":"we take the inner product of f with g and make the corresponding modifications."},{"Start":"04:16.270 ","End":"04:24.320","Text":"I didn\u0027t say it, but the C_n with capital C is the corresponding coefficients for g,"},{"Start":"04:24.320 ","End":"04:26.165","Text":"just like c_n belongs to f,"},{"Start":"04:26.165 ","End":"04:31.690","Text":"big C_n belongs to g. I\u0027ve color-coded it to make it clearer."},{"Start":"04:31.690 ","End":"04:34.665","Text":"In the case of the reals,"},{"Start":"04:34.665 ","End":"04:36.830","Text":"we have exactly a similar thing."},{"Start":"04:36.830 ","End":"04:39.230","Text":"Instead of the absolute value squared,"},{"Start":"04:39.230 ","End":"04:41.960","Text":"we have the product of each one with"},{"Start":"04:41.960 ","End":"04:46.525","Text":"its conjugate of the corresponding coefficient here, here, and here."},{"Start":"04:46.525 ","End":"04:50.895","Text":"Next we have Term-by-term differentiation."},{"Start":"04:50.895 ","End":"04:56.000","Text":"The condition is practically the same as the condition on"},{"Start":"04:56.000 ","End":"05:03.005","Text":"uniform convergence that f has to be continuous on the interval equal at the endpoints."},{"Start":"05:03.005 ","End":"05:07.370","Text":"It has to be piecewise continuously differentiable,"},{"Start":"05:07.370 ","End":"05:10.645","Text":"or at least f prime has to be piecewise continuous."},{"Start":"05:10.645 ","End":"05:15.020","Text":"In this case, then you can differentiate the Fourier series term by term,"},{"Start":"05:15.020 ","End":"05:20.180","Text":"and you get the Fourier series for f prime as exactly the derivatives term"},{"Start":"05:20.180 ","End":"05:25.805","Text":"wise of the case for f. This is how it looks in the case of the reals."},{"Start":"05:25.805 ","End":"05:28.850","Text":"This is in the case of the complex."},{"Start":"05:28.850 ","End":"05:35.080","Text":"As always, we have this 2Pi over b minus a factor to scale everything."},{"Start":"05:35.080 ","End":"05:39.300","Text":"Now integration, if f is in E(a,"},{"Start":"05:39.300 ","End":"05:42.645","Text":"b) and the Fourier series for f can be"},{"Start":"05:42.645 ","End":"05:47.760","Text":"integrated term-by-term to get that for all x in a,"},{"Start":"05:47.760 ","End":"05:52.320","Text":"b, the real version looks a real mess,"},{"Start":"05:52.320 ","End":"05:58.385","Text":"if you notice everywhere the 2Pi over b minus a or b minus a over 2Pi."},{"Start":"05:58.385 ","End":"06:00.830","Text":"Anyway, this is what it is in the real case."},{"Start":"06:00.830 ","End":"06:02.480","Text":"This is what it is in the complex case."},{"Start":"06:02.480 ","End":"06:05.255","Text":"Notice that I\u0027ve written equals,"},{"Start":"06:05.255 ","End":"06:07.115","Text":"not just the Tilde,"},{"Start":"06:07.115 ","End":"06:08.645","Text":"because it is actually equal,"},{"Start":"06:08.645 ","End":"06:15.605","Text":"because the series converges uniformly in this case to the function on the left."},{"Start":"06:15.605 ","End":"06:19.270","Text":"That\u0027s basically it for the theory part."},{"Start":"06:19.270 ","End":"06:21.740","Text":"Let\u0027s just do an exercise."},{"Start":"06:21.740 ","End":"06:24.770","Text":"For the exercise, why don\u0027t we just compute"},{"Start":"06:24.770 ","End":"06:30.210","Text":"the Fourier series for a function that\u0027s not on minus Pi, Pi."},{"Start":"06:30.210 ","End":"06:32.900","Text":"In this case, we\u0027ll take it from 0 to 2."},{"Start":"06:32.900 ","End":"06:34.879","Text":"I\u0027ll remind you of the formula."},{"Start":"06:34.879 ","End":"06:38.665","Text":"We\u0027ll do the complex case as requested."},{"Start":"06:38.665 ","End":"06:45.215","Text":"This is the formula for the Fourier series where the coefficients are given as follows."},{"Start":"06:45.215 ","End":"06:48.700","Text":"In our case, a is 0 and b is 2,"},{"Start":"06:48.700 ","End":"06:53.655","Text":"note that the 2 cancels with 2 minus 0."},{"Start":"06:53.655 ","End":"06:56.790","Text":"2 over b minus a is 1,"},{"Start":"06:56.790 ","End":"07:00.690","Text":"so we just have the nPi x,"},{"Start":"07:00.690 ","End":"07:04.365","Text":"f(x) is e^x over 2."},{"Start":"07:04.365 ","End":"07:06.225","Text":"Again, b minus a is 2."},{"Start":"07:06.225 ","End":"07:11.070","Text":"This is what we have to compute and this is equal 2."},{"Start":"07:11.070 ","End":"07:13.400","Text":"We can combine the exponents."},{"Start":"07:13.400 ","End":"07:18.190","Text":"This is 1/2 minus this and put it over a common denominator 2."},{"Start":"07:18.190 ","End":"07:23.045","Text":"Then we can divide by the coefficient of x,"},{"Start":"07:23.045 ","End":"07:27.170","Text":"which means multiplying by the upside down fraction."},{"Start":"07:27.170 ","End":"07:30.770","Text":"Then it\u0027s just the same thing with the e to the power of."},{"Start":"07:30.770 ","End":"07:34.665","Text":"Then plug in 0 and 2 and subtract."},{"Start":"07:34.665 ","End":"07:37.965","Text":"We get, if you put 2 in,"},{"Start":"07:37.965 ","End":"07:40.590","Text":"then x over 2 is 2 over 2 is 1."},{"Start":"07:40.590 ","End":"07:43.170","Text":"We just have e to this part."},{"Start":"07:43.170 ","End":"07:47.010","Text":"If you put in 0, it\u0027s e^0 which is 1."},{"Start":"07:47.010 ","End":"07:49.560","Text":"Also, the 2 with this 2 cancel."},{"Start":"07:49.560 ","End":"07:52.890","Text":"So we have this and this is equal to,"},{"Start":"07:52.890 ","End":"07:58.870","Text":"we can split this up into e times e to the minus 2i nPi."},{"Start":"07:58.870 ","End":"08:05.615","Text":"Now, each of the power of 2Pi i times any integer positive or negative is 1."},{"Start":"08:05.615 ","End":"08:08.640","Text":"This is just e times 1,"},{"Start":"08:08.640 ","End":"08:11.040","Text":"which is e minus 1 over this."},{"Start":"08:11.040 ","End":"08:14.380","Text":"We\u0027ll get rid of the imaginary part of the denominator by"},{"Start":"08:14.380 ","End":"08:17.860","Text":"multiplying top and bottom by the conjugate."},{"Start":"08:17.860 ","End":"08:19.405","Text":"If we do that,"},{"Start":"08:19.405 ","End":"08:22.990","Text":"what we\u0027re left with is this."},{"Start":"08:22.990 ","End":"08:25.660","Text":"That\u0027s about as simplified as it gets."},{"Start":"08:25.660 ","End":"08:27.175","Text":"Not too bad."},{"Start":"08:27.175 ","End":"08:32.080","Text":"Of course, now we just have to put it into the formula for the Fourier series."},{"Start":"08:32.080 ","End":"08:33.625","Text":"Now that we have the coefficient,"},{"Start":"08:33.625 ","End":"08:35.055","Text":"all this is c_n,"},{"Start":"08:35.055 ","End":"08:41.400","Text":"we have that the Fourier series for f is the following. We are done."}],"ID":28789},{"Watched":false,"Name":"Exercise 1","Duration":"3m 19s","ChapterTopicVideoID":27580,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.460","Text":"In this exercise, we\u0027re given the function f(x) =x^2 on the interval"},{"Start":"00:05.460 ","End":"00:10.680","Text":"from 0-2Pi and we have to find its real Fourier series."},{"Start":"00:10.680 ","End":"00:15.435","Text":"Here\u0027s a sketch of what it looks like when we make it periodic and it\u0027s not important."},{"Start":"00:15.435 ","End":"00:21.720","Text":"Now the formula for Fourier series on a general interval is this,"},{"Start":"00:21.720 ","End":"00:26.745","Text":"where a_naught is given by this formula, a_n and b_n."},{"Start":"00:26.745 ","End":"00:31.810","Text":"In our case, a is 0 and b is 2Pi."},{"Start":"00:31.850 ","End":"00:35.925","Text":"2 over b minus a is 1 over Pi."},{"Start":"00:35.925 ","End":"00:39.025","Text":"In this expression here is nx."},{"Start":"00:39.025 ","End":"00:41.615","Text":"What we get in our case,"},{"Start":"00:41.615 ","End":"00:47.060","Text":"all these formulas translated into this particular instance gives us follows."},{"Start":"00:47.060 ","End":"00:48.830","Text":"Now we\u0027re going to compute a_naught,"},{"Start":"00:48.830 ","End":"00:50.510","Text":"a_n, and b_n."},{"Start":"00:50.510 ","End":"00:54.315","Text":"A_n is the integral of f(x),"},{"Start":"00:54.315 ","End":"00:59.295","Text":"which is x^2 cosine nx dx from 0-2Pi."},{"Start":"00:59.295 ","End":"01:00.960","Text":"We\u0027ll calculate this in a moment."},{"Start":"01:00.960 ","End":"01:04.735","Text":"While we\u0027re at it, let\u0027s write down the formula for a_naught."},{"Start":"01:04.735 ","End":"01:06.140","Text":"Actually, that\u0027s easy to do."},{"Start":"01:06.140 ","End":"01:11.350","Text":"It\u0027s the integral, just x^2 from 0-2Pi and 1/Pi times that."},{"Start":"01:11.350 ","End":"01:13.420","Text":"It comes out 8Pi^2/3."},{"Start":"01:13.420 ","End":"01:14.650","Text":"That\u0027s the easy one."},{"Start":"01:14.650 ","End":"01:16.820","Text":"Let\u0027s continue with a_n,"},{"Start":"01:16.820 ","End":"01:18.300","Text":"this is equal to,"},{"Start":"01:18.300 ","End":"01:20.400","Text":"do an integration by parts."},{"Start":"01:20.400 ","End":"01:24.050","Text":"Think of this as fg prime. We need fg."},{"Start":"01:24.050 ","End":"01:26.780","Text":"The integral of cosine x is sine nx over"},{"Start":"01:26.780 ","End":"01:34.190","Text":"n minus the integral of the derivative of this times this, again."},{"Start":"01:34.190 ","End":"01:39.815","Text":"This part is 0. If you substitute 0 or 2 Pi, we get 0."},{"Start":"01:39.815 ","End":"01:42.505","Text":"We want the integral of that."},{"Start":"01:42.505 ","End":"01:46.170","Text":"Bring out the 2 over n in front minus 2 over nPi,"},{"Start":"01:46.170 ","End":"01:48.810","Text":"the integral of x sine x dx."},{"Start":"01:48.810 ","End":"01:54.035","Text":"Again by parts, take x times the integral of sine x,"},{"Start":"01:54.035 ","End":"01:57.890","Text":"which is minus cosine nx over n minus which makes it a"},{"Start":"01:57.890 ","End":"02:04.100","Text":"plus the derivative of x and cosine nx over n, as is."},{"Start":"02:04.100 ","End":"02:08.075","Text":"This part comes out to be 0 because"},{"Start":"02:08.075 ","End":"02:14.330","Text":"cosine of n times 2Pi is the same as cosine of n times 0."},{"Start":"02:14.330 ","End":"02:16.490","Text":"Each case it\u0027s cosine 0, which is 1."},{"Start":"02:16.490 ","End":"02:18.815","Text":"We get the same thing minus itself."},{"Start":"02:18.815 ","End":"02:21.620","Text":"This part is 0 and this part,"},{"Start":"02:21.620 ","End":"02:24.515","Text":"well the minus cancels with the minus,"},{"Start":"02:24.515 ","End":"02:30.035","Text":"we plug in 2Pi, we get cosine 2nPi and here 2Pi."},{"Start":"02:30.035 ","End":"02:31.880","Text":"If you plug in 0, well,"},{"Start":"02:31.880 ","End":"02:34.085","Text":"it doesn\u0027t matter, it\u0027s 0 times something."},{"Start":"02:34.085 ","End":"02:35.525","Text":"This is what we have,"},{"Start":"02:35.525 ","End":"02:41.315","Text":"and this boils down to a_n equals 4^2."},{"Start":"02:41.315 ","End":"02:44.695","Text":"B_n is this."},{"Start":"02:44.695 ","End":"02:47.585","Text":"This is equal to again by parts."},{"Start":"02:47.585 ","End":"02:49.910","Text":"You know what, I\u0027ll just leave it written out here."},{"Start":"02:49.910 ","End":"02:52.210","Text":"It\u0027s very similar to the other one."},{"Start":"02:52.210 ","End":"02:57.315","Text":"B_n comes out to be minus 4Pi/."},{"Start":"02:57.315 ","End":"03:01.140","Text":"Just to remember, a_naught was this,"},{"Start":"03:01.140 ","End":"03:03.745","Text":"and now we have a_n, b_n and a_naught."},{"Start":"03:03.745 ","End":"03:07.740","Text":"Now we can plug it into this formula which we had."},{"Start":"03:07.740 ","End":"03:10.920","Text":"It comes out to be this."},{"Start":"03:10.920 ","End":"03:17.630","Text":"This is our Fourier series for x^2 on the interval from 0-2Pi."},{"Start":"03:17.630 ","End":"03:20.520","Text":"That concludes this exercise."}],"ID":28790},{"Watched":false,"Name":"Exercise 2 - Part a","Duration":"5m 12s","ChapterTopicVideoID":27581,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:05.310","Text":"In this exercise, f(x) is the function minimum of 1 and absolute value"},{"Start":"00:05.310 ","End":"00:10.080","Text":"of x on the interval minus 2-2 as shown in the sketch here."},{"Start":"00:10.080 ","End":"00:12.120","Text":"In Part a, we have to find the Fourier"},{"Start":"00:12.120 ","End":"00:16.860","Text":"coefficients a_n and b_n of the real Fourier series of f,"},{"Start":"00:16.860 ","End":"00:21.990","Text":"and in Part b we have to compute these series."},{"Start":"00:21.990 ","End":"00:24.165","Text":"We\u0027ll start with Part a."},{"Start":"00:24.165 ","End":"00:28.875","Text":"The general formula for real Fourier series is this,"},{"Start":"00:28.875 ","End":"00:32.845","Text":"and the coefficients are given by these formulas."},{"Start":"00:32.845 ","End":"00:36.845","Text":"In our case, a is minus 2 and b is 2,"},{"Start":"00:36.845 ","End":"00:45.740","Text":"so 2 over b minus a is 1/2 and 2nPix over b minus a is nPix over 2."},{"Start":"00:45.740 ","End":"00:53.370","Text":"If we plug all that in then we get this formula for f(x) as a Fourier series,"},{"Start":"00:53.370 ","End":"00:58.190","Text":"and these are the formulas for a_naught, a_n, and b_n."},{"Start":"00:58.190 ","End":"00:59.870","Text":"Let\u0027s compute each of these."},{"Start":"00:59.870 ","End":"01:03.545","Text":"First note that f is an even function."},{"Start":"01:03.545 ","End":"01:09.340","Text":"For a_naught and a_n we can just double the integral from 0-2,"},{"Start":"01:09.340 ","End":"01:13.820","Text":"and for b_n it\u0027s going to be 0 because this is even"},{"Start":"01:13.820 ","End":"01:18.830","Text":"times odd is odd and an odd function on a symmetric interval gives 0."},{"Start":"01:18.830 ","End":"01:21.380","Text":"We just need to compute a_naught and a_n."},{"Start":"01:21.380 ","End":"01:23.050","Text":"Start with a_naught."},{"Start":"01:23.050 ","End":"01:26.385","Text":"We can break the integral up from 0-2"},{"Start":"01:26.385 ","End":"01:30.300","Text":"and the integral from 0-1 plus the integral from 1-2."},{"Start":"01:30.300 ","End":"01:31.875","Text":"Here it\u0027s equal to x,"},{"Start":"01:31.875 ","End":"01:33.330","Text":"here it\u0027s equal to 1,"},{"Start":"01:33.330 ","End":"01:36.015","Text":"so we get 3 over 2."},{"Start":"01:36.015 ","End":"01:38.715","Text":"For a_n similarly,"},{"Start":"01:38.715 ","End":"01:42.990","Text":"this part f(x) is x and this part f(x) is 1."},{"Start":"01:42.990 ","End":"01:45.875","Text":"We need to add these 2 integrals."},{"Start":"01:45.875 ","End":"01:48.620","Text":"This one will do by parts."},{"Start":"01:48.620 ","End":"01:51.115","Text":"This is f and this is g prime."},{"Start":"01:51.115 ","End":"01:54.770","Text":"Here\u0027s f and here\u0027s g. The integral"},{"Start":"01:54.770 ","End":"01:57.620","Text":"of cosine is the sine divided by the inner derivative,"},{"Start":"01:57.620 ","End":"02:04.550","Text":"so it\u0027s 2 over nPi here minus the integral of f prime g. F prime is 1,"},{"Start":"02:04.550 ","End":"02:06.120","Text":"g just like here;"},{"Start":"02:06.120 ","End":"02:09.370","Text":"2 over nPi sine nPix over 2,"},{"Start":"02:09.370 ","End":"02:11.578","Text":"and then this is the integral of the last part."},{"Start":"02:11.578 ","End":"02:13.781","Text":"The integral of cosine is sine,"},{"Start":"02:13.781 ","End":"02:16.580","Text":"we have to divide by nPi over 2."},{"Start":"02:16.580 ","End":"02:20.525","Text":"Now notice when x equals 1 here,"},{"Start":"02:20.525 ","End":"02:26.390","Text":"we get 2 over nPi sine of nPi over 2 when."},{"Start":"02:26.390 ","End":"02:29.195","Text":"We put x equals 1 in the last term,"},{"Start":"02:29.195 ","End":"02:32.770","Text":"we also get 2 over nPi sine nPi over 2."},{"Start":"02:32.770 ","End":"02:35.060","Text":"This part cancels with this part."},{"Start":"02:35.060 ","End":"02:36.470","Text":"Something else cancels."},{"Start":"02:36.470 ","End":"02:40.235","Text":"If you put in x equals 0 here, we get 0."},{"Start":"02:40.235 ","End":"02:44.230","Text":"If we put x equals 2 here,"},{"Start":"02:44.230 ","End":"02:52.295","Text":"this part is 2nPi over 2 which is nPi and sine of nPi is 0."},{"Start":"02:52.295 ","End":"02:54.335","Text":"That\u0027s 0 and that\u0027s 0."},{"Start":"02:54.335 ","End":"02:58.280","Text":"These two disappear and we\u0027re just left with this middle term."},{"Start":"02:58.280 ","End":"03:00.290","Text":"Now, let\u0027s do this one."},{"Start":"03:00.290 ","End":"03:03.440","Text":"The integral of sine is minus cosine,"},{"Start":"03:03.440 ","End":"03:06.805","Text":"or you could say the integral of minus sine is cosine."},{"Start":"03:06.805 ","End":"03:10.130","Text":"We have nPi over x and we also have to divide by"},{"Start":"03:10.130 ","End":"03:13.704","Text":"nPi over 2 which is 2 multiply by 2 over nPi,"},{"Start":"03:13.704 ","End":"03:16.250","Text":"but we already have 2 over nPi."},{"Start":"03:16.250 ","End":"03:17.935","Text":"So it\u0027s squared."},{"Start":"03:17.935 ","End":"03:20.525","Text":"Now when x equals 1,"},{"Start":"03:20.525 ","End":"03:23.420","Text":"we get cosine nPi over 2."},{"Start":"03:23.420 ","End":"03:26.370","Text":"When x is 0 we get cosine of 0 which is 1,"},{"Start":"03:26.370 ","End":"03:29.210","Text":"so this is what we have here."},{"Start":"03:29.210 ","End":"03:33.185","Text":"Continuing, this is what we have for a_n."},{"Start":"03:33.185 ","End":"03:35.750","Text":"If you plug in different values of n,"},{"Start":"03:35.750 ","End":"03:37.010","Text":"n equals 1, 2, 3,"},{"Start":"03:37.010 ","End":"03:38.090","Text":"4, 5,"},{"Start":"03:38.090 ","End":"03:39.605","Text":"6, 7,8,"},{"Start":"03:39.605 ","End":"03:43.025","Text":"then it cycles in the period of 4."},{"Start":"03:43.025 ","End":"03:44.510","Text":"Every 4 it repeats."},{"Start":"03:44.510 ","End":"03:49.050","Text":"We get cosine of Pi over 2 is 0,"},{"Start":"03:49.050 ","End":"03:50.790","Text":"cosine of Pi is minus 1,"},{"Start":"03:50.790 ","End":"03:52.470","Text":"cosine 3 Pi over 2,"},{"Start":"03:52.470 ","End":"03:54.420","Text":"and cosine 2Pi,"},{"Start":"03:54.420 ","End":"03:55.725","Text":"and then it repeats."},{"Start":"03:55.725 ","End":"04:00.980","Text":"We can break it up into 4 cases depending on whether n is 4k minus 3,"},{"Start":"04:00.980 ","End":"04:02.060","Text":"minus 2, minus 1,"},{"Start":"04:02.060 ","End":"04:05.350","Text":"or an even 4k and we get these."},{"Start":"04:05.350 ","End":"04:08.390","Text":"I colored it so you can see what belongs to what."},{"Start":"04:08.390 ","End":"04:10.894","Text":"This case and this case can be combined."},{"Start":"04:10.894 ","End":"04:13.502","Text":"Altogether they give the odd numbers,"},{"Start":"04:13.502 ","End":"04:20.780","Text":"so we can write it as a_n is this for n equals 2k minus 1 and then here we just repeat."},{"Start":"04:20.780 ","End":"04:23.065","Text":"This is for 4k minus 2,"},{"Start":"04:23.065 ","End":"04:28.015","Text":"and the last case we can write as otherwise. That\u0027s a_n."},{"Start":"04:28.015 ","End":"04:32.195","Text":"Let\u0027s remember what are a 0 and b_n."},{"Start":"04:32.195 ","End":"04:33.695","Text":"Now that we have these,"},{"Start":"04:33.695 ","End":"04:38.080","Text":"we can plug them into the Fourier series for f(x)."},{"Start":"04:38.080 ","End":"04:40.110","Text":"Of course the b_n is 0."},{"Start":"04:40.110 ","End":"04:41.477","Text":"This part we don\u0027t need,"},{"Start":"04:41.477 ","End":"04:45.395","Text":"so what are we left with is a_naught over 2 is 3/4."},{"Start":"04:45.395 ","End":"04:51.680","Text":"This has 3 parts because the 0 part we don\u0027t have to take into consideration."},{"Start":"04:51.680 ","End":"04:57.047","Text":"The bit for 2k minus 1 is minus 4 over n^2 Pi^2 squared,"},{"Start":"04:57.047 ","End":"05:00.200","Text":"but n is 2k minus 1 in this case."},{"Start":"05:00.200 ","End":"05:05.960","Text":"Similarly here we have minus e8 and here n is 4k minus 2."},{"Start":"05:05.960 ","End":"05:09.275","Text":"This is what we have for the Fourier series,"},{"Start":"05:09.275 ","End":"05:12.480","Text":"and that concludes Part a."}],"ID":28791},{"Watched":false,"Name":"Exercise 2 - Part b","Duration":"6m 23s","ChapterTopicVideoID":27582,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.565","Text":"Now we come to Part B of this exercise where we have to compute these series."},{"Start":"00:05.565 ","End":"00:08.610","Text":"Just remember the result of Part A."},{"Start":"00:08.610 ","End":"00:15.310","Text":"We had the expansion of f as a Fourier series was like this."},{"Start":"00:15.320 ","End":"00:19.080","Text":"We\u0027re going to use Dirichlet theorem also"},{"Start":"00:19.080 ","End":"00:23.130","Text":"adapted for the general Fourier series and a general interval."},{"Start":"00:23.130 ","End":"00:24.600","Text":"This is what it says."},{"Start":"00:24.600 ","End":"00:30.235","Text":"It\u0027s the same as the regular one except for the 2b and a."},{"Start":"00:30.235 ","End":"00:35.945","Text":"We can apply this theorem to our f because it is piecewise continuous."},{"Start":"00:35.945 ","End":"00:37.940","Text":"It\u0027s even continuous."},{"Start":"00:37.940 ","End":"00:40.955","Text":"At the point where x equals naught,"},{"Start":"00:40.955 ","End":"00:43.580","Text":"it does have a right-hand limit of the"},{"Start":"00:43.580 ","End":"00:46.550","Text":"derivative and the left-hand limit of the derivative."},{"Start":"00:46.550 ","End":"00:47.840","Text":"There may not be the same,"},{"Start":"00:47.840 ","End":"00:49.955","Text":"but they both exist, which is all we need."},{"Start":"00:49.955 ","End":"00:54.455","Text":"By the theorem, we get that the sum of the series"},{"Start":"00:54.455 ","End":"01:00.770","Text":"is the average of f from the right and f from the left."},{"Start":"01:00.770 ","End":"01:02.660","Text":"Well, they\u0027re both the same in this case."},{"Start":"01:02.660 ","End":"01:06.050","Text":"They\u0027re both 0 and when they\u0027re the same, it\u0027s just f(x)."},{"Start":"01:06.050 ","End":"01:09.200","Text":"That\u0027s what you get when f is continuous at a point."},{"Start":"01:09.200 ","End":"01:17.260","Text":"In our case, it comes out to be this f(x) is 0 and a naught is 3/2,"},{"Start":"01:17.260 ","End":"01:19.670","Text":"so a naught over 2 is 3/4,"},{"Start":"01:19.670 ","End":"01:23.045","Text":"and a_n was equal to this."},{"Start":"01:23.045 ","End":"01:27.349","Text":"When x is 0, this is also 0,"},{"Start":"01:27.349 ","End":"01:29.930","Text":"so cosine of this is 1."},{"Start":"01:29.930 ","End":"01:32.375","Text":"We just have the a_n times this."},{"Start":"01:32.375 ","End":"01:38.149","Text":"Similarly here, a_n was in two parts."},{"Start":"01:38.149 ","End":"01:44.840","Text":"The second part is minus 8 over Pi^2 4k minus 2^2."},{"Start":"01:44.840 ","End":"01:47.990","Text":"The b_n is a 0, so this doesn\u0027t participate."},{"Start":"01:47.990 ","End":"01:50.190","Text":"Just that this is twice."},{"Start":"01:50.190 ","End":"01:55.215","Text":"Notice that this 4k minus 2 is twice 2k minus 1."},{"Start":"01:55.215 ","End":"01:57.735","Text":"When it\u0027s squared, this is 4 times this."},{"Start":"01:57.735 ","End":"02:04.645","Text":"We can divide top and bottom by 4 here and get minus 2 here and just 2k minus 1 here."},{"Start":"02:04.645 ","End":"02:06.695","Text":"Reason to do this is, is that now,"},{"Start":"02:06.695 ","End":"02:12.450","Text":"these are of the same kind and we can combine them minus 4 and minus 2 is minus 6,"},{"Start":"02:12.450 ","End":"02:15.400","Text":"which we can bring over to the other side and call it 6."},{"Start":"02:15.400 ","End":"02:19.405","Text":"We have sum of 6 over this is 3/4."},{"Start":"02:19.405 ","End":"02:26.190","Text":"Now we can bring the 6 over pi^2 to the other side and get 3/4 times Pi^2 over 6,"},{"Start":"02:26.190 ","End":"02:28.395","Text":"which is Pi^2 over 8."},{"Start":"02:28.395 ","End":"02:32.990","Text":"We\u0027re almost done, not quite because we had something similar to"},{"Start":"02:32.990 ","End":"02:37.765","Text":"show K goes from 0 to infinity and 2k plus 1."},{"Start":"02:37.765 ","End":"02:41.225","Text":"But that\u0027s fixed by a simple change of index."},{"Start":"02:41.225 ","End":"02:49.285","Text":"If we take k from 0 to infinity and compensate by replacing k with k plus 1."},{"Start":"02:49.285 ","End":"02:51.510","Text":"Then as k goes 0,1,2,"},{"Start":"02:51.510 ","End":"02:54.560","Text":"et cetera, k plus 1 goes 1,2,3, et cetera."},{"Start":"02:54.560 ","End":"02:56.420","Text":"Just like we had here."},{"Start":"02:56.420 ","End":"03:00.240","Text":"Replacing that, and then simplifying this,"},{"Start":"03:00.240 ","End":"03:03.370","Text":"2k plus 1 minus 1 is 2k plus 1."},{"Start":"03:03.370 ","End":"03:06.565","Text":"Now we get exactly in the form that we wanted,"},{"Start":"03:06.565 ","End":"03:11.120","Text":"and Pi^2 over 8 is the answer for the first series."},{"Start":"03:11.120 ","End":"03:19.240","Text":"Reminder, f(x) is this on this interval and Fourier series for f(x) is this."},{"Start":"03:19.240 ","End":"03:22.745","Text":"Now we\u0027re going to use Parseval\u0027s identity."},{"Start":"03:22.745 ","End":"03:26.165","Text":"This is what it says for a generalized interval,"},{"Start":"03:26.165 ","End":"03:31.160","Text":"we have a 2 over b minus a integral from a to b of f(x) squared"},{"Start":"03:31.160 ","End":"03:36.380","Text":"and is a sum of all the squares of the coefficients except for the a naught part,"},{"Start":"03:36.380 ","End":"03:37.940","Text":"which is slightly different."},{"Start":"03:37.940 ","End":"03:40.340","Text":"Anyway, in our case,"},{"Start":"03:40.340 ","End":"03:43.850","Text":"we just evaluate just the left-hand side."},{"Start":"03:43.850 ","End":"03:50.870","Text":"This is equal to 1.5 integral of absolute value of f of x squared."},{"Start":"03:50.870 ","End":"03:53.090","Text":"Because of the evenness,"},{"Start":"03:53.090 ","End":"03:57.935","Text":"we can double it and just take it from 0 to 2."},{"Start":"03:57.935 ","End":"04:02.585","Text":"F(x) from 0 to 2 is x squared."},{"Start":"04:02.585 ","End":"04:08.255","Text":"The integral from 0 to 2 is the integral from 0 to 1 plus from 1 to 2."},{"Start":"04:08.255 ","End":"04:10.580","Text":"Here, f(x) equals x,"},{"Start":"04:10.580 ","End":"04:12.060","Text":"here f(x) equals 1,"},{"Start":"04:12.060 ","End":"04:13.985","Text":"so this is what we get."},{"Start":"04:13.985 ","End":"04:18.710","Text":"This is equal to 4/3."},{"Start":"04:18.710 ","End":"04:20.510","Text":"That was the left-hand side here."},{"Start":"04:20.510 ","End":"04:22.415","Text":"Now the right-hand side,"},{"Start":"04:22.415 ","End":"04:24.755","Text":"we can just plug it all in."},{"Start":"04:24.755 ","End":"04:29.305","Text":"A naught is 3/2."},{"Start":"04:29.305 ","End":"04:33.240","Text":"We get absolute value of 3/2^ over 2,"},{"Start":"04:33.240 ","End":"04:36.390","Text":"and then a_n is this plus this."},{"Start":"04:36.390 ","End":"04:38.240","Text":"The b_n\u0027s are nothing."},{"Start":"04:38.240 ","End":"04:40.250","Text":"This disappears, but a_n,"},{"Start":"04:40.250 ","End":"04:43.100","Text":"has 2 sets of indices."},{"Start":"04:43.100 ","End":"04:46.010","Text":"The 4/3 is from here of course."},{"Start":"04:46.010 ","End":"04:47.825","Text":"Let\u0027s see what we get."},{"Start":"04:47.825 ","End":"04:52.355","Text":"This is 9/4 divided by 2 is 9/8."},{"Start":"04:52.355 ","End":"04:57.355","Text":"Here we have 16 over Pi^4, 2k minus 1^4."},{"Start":"04:57.355 ","End":"05:01.200","Text":"Here, 8^2 is 64,"},{"Start":"05:01.200 ","End":"05:04.625","Text":"Pi^4 4k minus 2^4."},{"Start":"05:04.625 ","End":"05:11.615","Text":"Just like before, we can simplify because 4k minus 2 is 2k minus 1 squared."},{"Start":"05:11.615 ","End":"05:14.840","Text":"We can divide top and bottom by 2^4,"},{"Start":"05:14.840 ","End":"05:18.230","Text":"which is 16, and get left with this."},{"Start":"05:18.230 ","End":"05:21.500","Text":"Also, we can bring the 9/8 to the other side."},{"Start":"05:21.500 ","End":"05:25.820","Text":"4/3 minus 9/8 is 5/24,"},{"Start":"05:25.820 ","End":"05:29.135","Text":"and here 16 and 4 is 20."},{"Start":"05:29.135 ","End":"05:34.040","Text":"Now we can take 20 over Pi^4 and bring it to the other side."},{"Start":"05:34.040 ","End":"05:36.799","Text":"We get this series was just a one here,"},{"Start":"05:36.799 ","End":"05:40.549","Text":"is 5/24 times Pi^4 over 20,"},{"Start":"05:40.549 ","End":"05:45.110","Text":"which comes out to be Pi^4 over 96."},{"Start":"05:45.110 ","End":"05:48.200","Text":"Now, we can make a change of index."},{"Start":"05:48.200 ","End":"05:52.025","Text":"We have to find the sum with a plus here."},{"Start":"05:52.025 ","End":"06:00.725","Text":"If we replace k with k plus 1 and take it just from 0 to infinity,"},{"Start":"06:00.725 ","End":"06:03.994","Text":"we\u0027ll get the same because when k goes from 0 to infinity,"},{"Start":"06:03.994 ","End":"06:07.175","Text":"k plus 1 goes from 1 to infinity."},{"Start":"06:07.175 ","End":"06:11.970","Text":"Twice k plus 1 minus 1 is 2k plus 1."},{"Start":"06:12.650 ","End":"06:17.080","Text":"This is the series we had to find."},{"Start":"06:17.090 ","End":"06:23.740","Text":"The answer is Pi^4 over 96, and we\u0027re done."}],"ID":28792},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"2m 18s","ChapterTopicVideoID":27583,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.965","Text":"In this exercise, we were asked to compute the Fourier series"},{"Start":"00:04.965 ","End":"00:10.675","Text":"for a function that\u0027s not on minus Pi Pi in this case from 0-2."},{"Start":"00:10.675 ","End":"00:12.630","Text":"I\u0027ll remind you of the formula."},{"Start":"00:12.630 ","End":"00:16.440","Text":"We\u0027ll do the complex case as requested."},{"Start":"00:16.440 ","End":"00:22.995","Text":"This is the formula for the Fourier series where the coefficients are given as follows."},{"Start":"00:22.995 ","End":"00:31.410","Text":"In our case, a is 0 and b is 2 note that the 2 cancels with 2 minus 0."},{"Start":"00:31.410 ","End":"00:34.650","Text":"2 over b minus a is 1."},{"Start":"00:34.650 ","End":"00:38.370","Text":"We just have the n Pi x,"},{"Start":"00:38.370 ","End":"00:42.135","Text":"f(x) is e^ x/2."},{"Start":"00:42.135 ","End":"00:43.980","Text":"Again, b minus a is 2."},{"Start":"00:43.980 ","End":"00:46.395","Text":"This is what we have to compute."},{"Start":"00:46.395 ","End":"00:48.840","Text":"This is equal to,"},{"Start":"00:48.840 ","End":"00:51.169","Text":"we can combine the exponents."},{"Start":"00:51.169 ","End":"00:55.960","Text":"This is 1/2 minus this and put it over a common denominator 2."},{"Start":"00:55.960 ","End":"01:00.815","Text":"Then we can divide by the coefficient of x,"},{"Start":"01:00.815 ","End":"01:04.940","Text":"which means multiplying by the upside-down fraction."},{"Start":"01:04.940 ","End":"01:08.555","Text":"Then it\u0027s just the same thing with the e to the power of,"},{"Start":"01:08.555 ","End":"01:12.425","Text":"then plug in 0 and 2 and subtract."},{"Start":"01:12.425 ","End":"01:15.725","Text":"We get, if you put 2 in,"},{"Start":"01:15.725 ","End":"01:18.335","Text":"then x/2 is 2/2 is 1."},{"Start":"01:18.335 ","End":"01:20.860","Text":"We just have e to this part."},{"Start":"01:20.860 ","End":"01:24.795","Text":"If you put in 0, it\u0027s e^0 which is 1."},{"Start":"01:24.795 ","End":"01:27.480","Text":"Also the 2 with this 2 cancel."},{"Start":"01:27.480 ","End":"01:30.670","Text":"We have this and this is equal 2,"},{"Start":"01:30.670 ","End":"01:36.635","Text":"we can split this up into e times e^-2in Pi."},{"Start":"01:36.635 ","End":"01:43.375","Text":"Now, each of the power of 2 Pi i times any integer positive or negative is 1."},{"Start":"01:43.375 ","End":"01:46.410","Text":"This is just e times 1,"},{"Start":"01:46.410 ","End":"01:48.810","Text":"which is e minus 1 over this."},{"Start":"01:48.810 ","End":"01:52.130","Text":"We\u0027ll get rid of the imaginary part of the denominator by"},{"Start":"01:52.130 ","End":"01:55.640","Text":"multiplying top and bottom by the conjugate."},{"Start":"01:55.640 ","End":"01:57.170","Text":"If we do that,"},{"Start":"01:57.170 ","End":"02:00.765","Text":"what we\u0027re left with is this."},{"Start":"02:00.765 ","End":"02:03.425","Text":"That\u0027s about as simplified as it gets."},{"Start":"02:03.425 ","End":"02:04.955","Text":"Not too bad."},{"Start":"02:04.955 ","End":"02:09.830","Text":"Of course, now we just have to put it into the formula for the Fourier series."},{"Start":"02:09.830 ","End":"02:11.405","Text":"Now that we have the co-efficient,"},{"Start":"02:11.405 ","End":"02:12.815","Text":"all this is cn."},{"Start":"02:12.815 ","End":"02:17.015","Text":"We have that the Fourier series for f is the following,"},{"Start":"02:17.015 ","End":"02:19.170","Text":"and we are done."}],"ID":28793},{"Watched":false,"Name":"Exercise 3 - Part b","Duration":"2m 1s","ChapterTopicVideoID":27584,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:03.420","Text":"Now, we come to part B of the exercise."},{"Start":"00:03.420 ","End":"00:04.560","Text":"In part a,"},{"Start":"00:04.560 ","End":"00:07.080","Text":"we have to find it complex Fourier series,"},{"Start":"00:07.080 ","End":"00:09.330","Text":"which turns out is this."},{"Start":"00:09.330 ","End":"00:13.770","Text":"Then part b, you want to find out what function the Fourier series"},{"Start":"00:13.770 ","End":"00:18.885","Text":"converges to and to sketch its graph at least 3 periods."},{"Start":"00:18.885 ","End":"00:28.380","Text":"Let f tilde be the periodic extension of f. Then it\u0027s continuous except at even integers."},{"Start":"00:28.380 ","End":"00:30.090","Text":"Between 0 and 2,"},{"Start":"00:30.090 ","End":"00:31.995","Text":"it\u0027s e to the x/2."},{"Start":"00:31.995 ","End":"00:35.830","Text":"Then we copy paste periodically."},{"Start":"00:36.170 ","End":"00:40.070","Text":"At the places where f tilde is continuous,"},{"Start":"00:40.070 ","End":"00:42.545","Text":"which is everywhere except for x equals 2n,"},{"Start":"00:42.545 ","End":"00:48.940","Text":"by Dirichlet\u0027s theorem, the series converges to f tilde, the periodic extension."},{"Start":"00:48.940 ","End":"00:51.105","Text":"At x equals 2n,"},{"Start":"00:51.105 ","End":"00:54.000","Text":"it converges to the average."},{"Start":"00:54.000 ","End":"00:58.730","Text":"As a jump discontinuity where we have the left limit and right limit,"},{"Start":"00:58.730 ","End":"01:00.155","Text":"but just not equal,"},{"Start":"01:00.155 ","End":"01:03.950","Text":"then it converges to the arithmetic mean of the 2."},{"Start":"01:03.950 ","End":"01:06.395","Text":"Let\u0027s take it at x equals 0,"},{"Start":"01:06.395 ","End":"01:07.655","Text":"because that\u0027s the easiest."},{"Start":"01:07.655 ","End":"01:10.085","Text":"Everywhere else it\u0027s the same because it\u0027s a periodic."},{"Start":"01:10.085 ","End":"01:12.800","Text":"Here on the left,"},{"Start":"01:12.800 ","End":"01:16.310","Text":"it\u0027s 0 minus, on the right 0 plus,"},{"Start":"01:16.310 ","End":"01:22.995","Text":"but here it\u0027s the same as 2 minus by the periodicity."},{"Start":"01:22.995 ","End":"01:27.350","Text":"Here it\u0027s equal to e to the power of 2 over 2."},{"Start":"01:27.350 ","End":"01:29.060","Text":"Just let x equals 2."},{"Start":"01:29.060 ","End":"01:31.550","Text":"Here, just let x equals 0."},{"Start":"01:31.550 ","End":"01:34.340","Text":"We get each of the 1 plus e to the 0 over 2."},{"Start":"01:34.340 ","End":"01:37.805","Text":"So E plus 1/2 is this value,"},{"Start":"01:37.805 ","End":"01:40.790","Text":"x equals an even number."},{"Start":"01:40.790 ","End":"01:45.500","Text":"The limit function for the series is equal to e"},{"Start":"01:45.500 ","End":"01:49.760","Text":"plus 1/2 when x is an even number and other than that,"},{"Start":"01:49.760 ","End":"01:53.090","Text":"when x is between an even number and the following even number,"},{"Start":"01:53.090 ","End":"01:55.100","Text":"we just subtract 2n from x."},{"Start":"01:55.100 ","End":"01:56.780","Text":"So it\u0027s between 0 and 2."},{"Start":"01:56.780 ","End":"01:59.735","Text":"Then e to the power of that over 2."},{"Start":"01:59.735 ","End":"02:02.640","Text":"That concludes Part B."}],"ID":28794},{"Watched":false,"Name":"Exercise 3 - Part c","Duration":"2m 48s","ChapterTopicVideoID":27585,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"Continuing with this exercise,"},{"Start":"00:02.460 ","End":"00:05.190","Text":"we\u0027ve done a and b and now we come to part c"},{"Start":"00:05.190 ","End":"00:08.385","Text":"where we have to find the sum of this series."},{"Start":"00:08.385 ","End":"00:14.565","Text":"In part a, the Fourier series we\u0027ve computed was this,"},{"Start":"00:14.565 ","End":"00:19.565","Text":"and in part b, what we need is that at x equals 0,"},{"Start":"00:19.565 ","End":"00:23.315","Text":"the series converges to e plus 1/2."},{"Start":"00:23.315 ","End":"00:30.090","Text":"In part c, what we can do is substitute x equals 0,"},{"Start":"00:30.090 ","End":"00:32.910","Text":"and then e^inx is 1,"},{"Start":"00:32.910 ","End":"00:34.965","Text":"so we just get this,"},{"Start":"00:34.965 ","End":"00:39.815","Text":"and we said that the series converges at 0 to e plus 1/2."},{"Start":"00:39.815 ","End":"00:42.560","Text":"e plus 1/2 equals the sum of this series."},{"Start":"00:42.560 ","End":"00:50.240","Text":"Bring the e minus 1 to the left-hand side and break it up into real and imaginary parts."},{"Start":"00:50.240 ","End":"00:53.150","Text":"There\u0027s an i here, 1 plus 2i something,"},{"Start":"00:53.150 ","End":"00:54.940","Text":"just the 1,"},{"Start":"00:54.940 ","End":"00:58.035","Text":"and then i times the 2n Pi."},{"Start":"00:58.035 ","End":"01:01.550","Text":"This is a split because the left-hand side is real,"},{"Start":"01:01.550 ","End":"01:04.250","Text":"it\u0027s equal to the real part of the right-hand side."},{"Start":"01:04.250 ","End":"01:06.155","Text":"We can say this equals this."},{"Start":"01:06.155 ","End":"01:07.460","Text":"There\u0027s an extra check."},{"Start":"01:07.460 ","End":"01:10.580","Text":"We can see that this sum is 0 because it goes from"},{"Start":"01:10.580 ","End":"01:14.110","Text":"minus infinity to infinity of each n as a minus n,"},{"Start":"01:14.110 ","End":"01:15.820","Text":"which cancels it out,"},{"Start":"01:15.820 ","End":"01:18.250","Text":"and when it\u0027s 0, it\u0027s equal to 0."},{"Start":"01:18.250 ","End":"01:23.795","Text":"It\u0027s completely symmetric and the plus part cancels the minus part, so this really is 0."},{"Start":"01:23.795 ","End":"01:26.075","Text":"This is getting close to what we want."},{"Start":"01:26.075 ","End":"01:27.890","Text":"We want from 1 to infinity."},{"Start":"01:27.890 ","End":"01:31.100","Text":"We can break this up into 3 parts."},{"Start":"01:31.100 ","End":"01:33.350","Text":"The part from 1 to infinity,"},{"Start":"01:33.350 ","End":"01:35.285","Text":"the value when n is 0,"},{"Start":"01:35.285 ","End":"01:38.840","Text":"and the part from minus infinity to minus 1."},{"Start":"01:38.840 ","End":"01:39.950","Text":"Now for the last part,"},{"Start":"01:39.950 ","End":"01:42.980","Text":"we can make a substitution change of index,"},{"Start":"01:42.980 ","End":"01:49.975","Text":"replace n with minus n and then n will go from 1 to infinity."},{"Start":"01:49.975 ","End":"01:54.050","Text":"We get this plus this and this now from 1 to"},{"Start":"01:54.050 ","End":"01:59.510","Text":"infinity and minus n^2 is the same as n^2. We get this."},{"Start":"01:59.510 ","End":"02:04.309","Text":"Now the first and the last part of this sum are equal."},{"Start":"02:04.309 ","End":"02:09.910","Text":"What we get is twice this and this is equal to 1."},{"Start":"02:09.910 ","End":"02:14.780","Text":"Now, let\u0027s bring the 1 to the other side and divide by 2,"},{"Start":"02:14.780 ","End":"02:17.495","Text":"so we get e plus 1/2 twice e minus 1,"},{"Start":"02:17.495 ","End":"02:21.290","Text":"takeaway 1 and divide it by 2 or multiply by a 1/2."},{"Start":"02:21.290 ","End":"02:22.930","Text":"This is equal 2,"},{"Start":"02:22.930 ","End":"02:24.575","Text":"bring this 2 a common denominator,"},{"Start":"02:24.575 ","End":"02:29.030","Text":"e plus 1 minus twice e minus 1 over twice e minus 1,"},{"Start":"02:29.030 ","End":"02:30.860","Text":"and then the 2 from here,"},{"Start":"02:30.860 ","End":"02:35.345","Text":"this comes out to be e minus 2e is minus e,"},{"Start":"02:35.345 ","End":"02:37.460","Text":"1 minus minus 2 is 3."},{"Start":"02:37.460 ","End":"02:42.845","Text":"3 minus e in the numerator and 4e minus 1 in the denominator."},{"Start":"02:42.845 ","End":"02:45.680","Text":"This is the sum of the series we want."},{"Start":"02:45.680 ","End":"02:48.840","Text":"That concludes this exercise."}],"ID":28795},{"Watched":false,"Name":"Exercise 4","Duration":"4m 8s","ChapterTopicVideoID":27586,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.740","Text":"In this exercise, we\u0027re going to find the Fourier expansion of the function"},{"Start":"00:04.740 ","End":"00:10.470","Text":"f(x) equals absolute value of x on the interval minus 1 to 1."},{"Start":"00:10.470 ","End":"00:13.710","Text":"We\u0027re going to use a generalized interval."},{"Start":"00:13.710 ","End":"00:14.969","Text":"These are the formulas."},{"Start":"00:14.969 ","End":"00:17.700","Text":"This is the Fourier expansion."},{"Start":"00:17.700 ","End":"00:21.480","Text":"Should say b and a are the endpoints of the interval,"},{"Start":"00:21.480 ","End":"00:23.505","Text":"and a note, a_n,"},{"Start":"00:23.505 ","End":"00:26.235","Text":"and b_n are given by these formulas."},{"Start":"00:26.235 ","End":"00:29.370","Text":"Let\u0027s see what we get in our case."},{"Start":"00:29.370 ","End":"00:31.455","Text":"A is minus 1,"},{"Start":"00:31.455 ","End":"00:34.125","Text":"b is 1, the endpoints of the interval."},{"Start":"00:34.125 ","End":"00:37.540","Text":"Then 2 over b minus a is 1,"},{"Start":"00:37.540 ","End":"00:39.770","Text":"and this here,"},{"Start":"00:39.770 ","End":"00:43.775","Text":"2nPix over b minus a reduces to nPix."},{"Start":"00:43.775 ","End":"00:46.370","Text":"Substitute all these here,"},{"Start":"00:46.370 ","End":"00:51.770","Text":"we get the Fourier expansion and the coefficients."},{"Start":"00:51.770 ","End":"00:54.620","Text":"Now we just have to compute a note,"},{"Start":"00:54.620 ","End":"00:57.800","Text":"a_n, and b_n and then substitute them in here."},{"Start":"00:57.800 ","End":"01:00.120","Text":"F is an even function,"},{"Start":"01:00.120 ","End":"01:03.740","Text":"so these 2 can be simplified by taking"},{"Start":"01:03.740 ","End":"01:07.854","Text":"the integral of just from 0-1 and then doubling here."},{"Start":"01:07.854 ","End":"01:11.990","Text":"Whereas b_n is 0 because f(x) is"},{"Start":"01:11.990 ","End":"01:14.420","Text":"an even function together with the sin that\u0027s"},{"Start":"01:14.420 ","End":"01:18.154","Text":"an odd function on a symmetric interval, so that\u0027s 0."},{"Start":"01:18.154 ","End":"01:21.805","Text":"Now we just have to compute a_0 and a_n."},{"Start":"01:21.805 ","End":"01:25.380","Text":"A_0 from this formula is the integral."},{"Start":"01:25.380 ","End":"01:29.160","Text":"The absolute value of x is just x from 0-1."},{"Start":"01:29.160 ","End":"01:34.475","Text":"This integral comes out to be 1/2 times 2 is 1."},{"Start":"01:34.475 ","End":"01:37.050","Text":"As for a_n, well,"},{"Start":"01:37.050 ","End":"01:39.875","Text":"we\u0027re going to do this as an integration by parts."},{"Start":"01:39.875 ","End":"01:45.840","Text":"We\u0027ll let the x be f and this cosine is g prime."},{"Start":"01:46.780 ","End":"01:51.750","Text":"G is the integral primitive of cosine,"},{"Start":"01:51.750 ","End":"01:59.465","Text":"which is sin and divide it by the derivative minus the integral of the x with the sin,"},{"Start":"01:59.465 ","End":"02:02.750","Text":"the fg Pi integral from 0-1."},{"Start":"02:02.750 ","End":"02:07.910","Text":"Now, this comes out to be 0 because when you plug in 1 or 0,"},{"Start":"02:07.910 ","End":"02:12.260","Text":"you get sin of multiples of Pi, which is 0."},{"Start":"02:12.260 ","End":"02:14.945","Text":"We\u0027re left the integral of this."},{"Start":"02:14.945 ","End":"02:18.535","Text":"The integral of minus sin is cosine,"},{"Start":"02:18.535 ","End":"02:21.460","Text":"so we have cosine of nPix,"},{"Start":"02:21.460 ","End":"02:23.330","Text":"but we divide by another nPi,"},{"Start":"02:23.330 ","End":"02:26.690","Text":"which makes it n^2 Pi^2 and there\u0027s a 2 from here."},{"Start":"02:26.690 ","End":"02:27.965","Text":"Plug in 1,"},{"Start":"02:27.965 ","End":"02:30.790","Text":"and we get cosine of n Pi,"},{"Start":"02:30.790 ","End":"02:32.480","Text":"which is minus 1^n."},{"Start":"02:32.480 ","End":"02:34.235","Text":"We\u0027ve seen this many times before,"},{"Start":"02:34.235 ","End":"02:36.320","Text":"plug in 0, we have 1."},{"Start":"02:36.320 ","End":"02:38.375","Text":"This is the expression we get."},{"Start":"02:38.375 ","End":"02:40.520","Text":"Now we have to separate odds and evens."},{"Start":"02:40.520 ","End":"02:41.735","Text":"If n is odd,"},{"Start":"02:41.735 ","End":"02:43.570","Text":"this is minus 1, if n is even,"},{"Start":"02:43.570 ","End":"02:45.560","Text":"this is plus 1."},{"Start":"02:45.560 ","End":"02:49.950","Text":"If it\u0027s even, we get 1 minus 1 is 0, and if it\u0027s odd,"},{"Start":"02:49.950 ","End":"02:54.540","Text":"minus 1, minus 1 is minus 2 together with the 2 gives minus 4."},{"Start":"02:54.540 ","End":"02:57.170","Text":"Replace n by 2k minus 1,"},{"Start":"02:57.170 ","End":"02:59.700","Text":"which is a typical general odd number."},{"Start":"02:59.700 ","End":"03:01.110","Text":"We have a_n,"},{"Start":"03:01.110 ","End":"03:05.520","Text":"and we also had a_0, and b_n."},{"Start":"03:05.520 ","End":"03:07.670","Text":"Let\u0027s summarize what we have so far."},{"Start":"03:07.670 ","End":"03:12.170","Text":"F(x) is the absolute value of x from minus 1 to 1."},{"Start":"03:12.170 ","End":"03:14.900","Text":"F(x) is the following series."},{"Start":"03:14.900 ","End":"03:17.435","Text":"But we found a_0, b_n,"},{"Start":"03:17.435 ","End":"03:20.990","Text":"and a_n, so we substitute all those in here."},{"Start":"03:20.990 ","End":"03:24.500","Text":"Also that f of x is absolute value of x so we get absolute value"},{"Start":"03:24.500 ","End":"03:27.995","Text":"of x is a_0 over 2 minus the sum."},{"Start":"03:27.995 ","End":"03:32.600","Text":"We take the sum over k. K goes from 1 to infinity,"},{"Start":"03:32.600 ","End":"03:35.035","Text":"gives us all the odd numbers,"},{"Start":"03:35.035 ","End":"03:39.305","Text":"and minus here, 4 here, and just copy."},{"Start":"03:39.305 ","End":"03:42.170","Text":"Then we have cosine nPix,"},{"Start":"03:42.170 ","End":"03:45.875","Text":"but n is 2k minus 1 so we get this,"},{"Start":"03:45.875 ","End":"03:48.670","Text":"and that is the answer."},{"Start":"03:48.670 ","End":"03:52.985","Text":"Just in case you want to know what this converges to,"},{"Start":"03:52.985 ","End":"03:59.360","Text":"it converges to the periodic extension of absolute value of x from minus 1 t21."},{"Start":"03:59.360 ","End":"04:04.400","Text":"We just replicate it to the whole real line, make it periodic."},{"Start":"04:04.400 ","End":"04:08.790","Text":"That\u0027s just a by the way. We\u0027re done."}],"ID":28796},{"Watched":false,"Name":"Exercise 5","Duration":"7m 39s","ChapterTopicVideoID":27587,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.450","Text":"In this exercise, f(x) is the following function defined from 0 to 2,"},{"Start":"00:06.450 ","End":"00:08.610","Text":"here it\u0027s equal to x,"},{"Start":"00:08.610 ","End":"00:11.415","Text":"and here it\u0027s equal to 2 minus x."},{"Start":"00:11.415 ","End":"00:13.350","Text":"If we want the sine-series,"},{"Start":"00:13.350 ","End":"00:19.110","Text":"we expand it to an odd function on a symmetric interval like so."},{"Start":"00:19.110 ","End":"00:22.455","Text":"There\u0027s a formula, this is what it is."},{"Start":"00:22.455 ","End":"00:25.785","Text":"This makes F, which is this whole thing,"},{"Start":"00:25.785 ","End":"00:31.670","Text":"an odd extension of f. F is equal to f on the part from 0-2."},{"Start":"00:31.670 ","End":"00:34.100","Text":"Now, because it\u0027s an odd function,"},{"Start":"00:34.100 ","End":"00:37.490","Text":"its Fourier expansion will only have sine series,"},{"Start":"00:37.490 ","End":"00:38.540","Text":"will only have b_ns,"},{"Start":"00:38.540 ","End":"00:40.954","Text":"all the a_ns will be zero."},{"Start":"00:40.954 ","End":"00:49.030","Text":"This will be the Fourier series in the general form with an interval from a to b."},{"Start":"00:49.030 ","End":"00:53.135","Text":"And this is what the coefficient b_n will be."},{"Start":"00:53.135 ","End":"00:58.610","Text":"Now, let\u0027s see what we can reduce it to because we know what a and b are;"},{"Start":"00:58.610 ","End":"01:01.655","Text":"minus 2 and 2."},{"Start":"01:01.655 ","End":"01:07.945","Text":"So this part, the 2 over b-a part becomes 1/2."},{"Start":"01:07.945 ","End":"01:12.265","Text":"This will become n Pi x over 2."},{"Start":"01:12.265 ","End":"01:15.890","Text":"These formulas, specifically in our case,"},{"Start":"01:15.890 ","End":"01:18.680","Text":"become these two formulas and now,"},{"Start":"01:18.680 ","End":"01:20.490","Text":"to compute b_n,"},{"Start":"01:20.490 ","End":"01:23.510","Text":"because it\u0027s an even function on a symmetric interval."},{"Start":"01:23.510 ","End":"01:26.030","Text":"Why even? This is odd and this is odd,"},{"Start":"01:26.030 ","End":"01:27.365","Text":"so together they are even,"},{"Start":"01:27.365 ","End":"01:32.290","Text":"then we can double the integral and just take it from 0 to 2."},{"Start":"01:32.290 ","End":"01:37.955","Text":"From 0 to 2, big F is the same as little f. So b_n is expressed like so in"},{"Start":"01:37.955 ","End":"01:43.175","Text":"terms of our original f and the remainder f(x) is this."},{"Start":"01:43.175 ","End":"01:49.470","Text":"Now, we have to compute this series of integrals for each n. Now,"},{"Start":"01:49.470 ","End":"01:51.510","Text":"we\u0027ll break it up into 2 parts;"},{"Start":"01:51.510 ","End":"01:53.450","Text":"from 0 to 1 and 1 to 2,"},{"Start":"01:53.450 ","End":"01:57.445","Text":"because this is defined separately from 0 to 1 and from 1 to 2."},{"Start":"01:57.445 ","End":"01:58.800","Text":"Between 0 and 1,"},{"Start":"01:58.800 ","End":"02:01.440","Text":"it\u0027s equal to x, between 1 and 2,"},{"Start":"02:01.440 ","End":"02:03.105","Text":"it\u0027s 2 minus x,"},{"Start":"02:03.105 ","End":"02:05.360","Text":"so we have to compute these two integrals."},{"Start":"02:05.360 ","End":"02:09.380","Text":"Each of these will do an integration by parts on."},{"Start":"02:09.380 ","End":"02:11.330","Text":"For the first one,"},{"Start":"02:11.330 ","End":"02:17.765","Text":"we\u0027ll take x as the g and this as the f\u0027."},{"Start":"02:17.765 ","End":"02:20.450","Text":"This is the one we have to integrate."},{"Start":"02:20.450 ","End":"02:25.140","Text":"The integral of this will be minus cosine,"},{"Start":"02:25.140 ","End":"02:28.410","Text":"that\u0027s not all, divided by n Pi over 2,"},{"Start":"02:28.410 ","End":"02:30.080","Text":"multiplied by 2 over n Pi."},{"Start":"02:30.080 ","End":"02:35.900","Text":"So this is the antiderivative or integral of this times this"},{"Start":"02:35.900 ","End":"02:43.665","Text":"minus the derivative of this times this integral from 0 to 1."},{"Start":"02:43.665 ","End":"02:46.995","Text":"Then similarly, for the other bit,"},{"Start":"02:46.995 ","End":"02:51.450","Text":"it\u0027s very similar, just 1 and 2 instead of 0 and 1."},{"Start":"02:51.450 ","End":"02:57.820","Text":"Here we have 2-x and here we have its derivative minus 1 and the rest of it is the same."},{"Start":"02:57.820 ","End":"03:01.170","Text":"Now, if we plug in x equals 1,"},{"Start":"03:01.170 ","End":"03:02.880","Text":"this x becomes a 1,"},{"Start":"03:02.880 ","End":"03:06.260","Text":"this x becomes a 1, this is what we have."},{"Start":"03:06.260 ","End":"03:10.430","Text":"But if we plug in 0, we get 0 because there\u0027s a 0 here."},{"Start":"03:10.430 ","End":"03:15.705","Text":"Minus, and the integral of this,"},{"Start":"03:15.705 ","End":"03:19.894","Text":"we have to take the integral of cosine, which is sine,"},{"Start":"03:19.894 ","End":"03:23.405","Text":"and then divide by n Pi over 2,"},{"Start":"03:23.405 ","End":"03:27.320","Text":"which gives this 2^2 over n^2 Pi squared,"},{"Start":"03:27.320 ","End":"03:32.195","Text":"and the minus here and minus here were there already."},{"Start":"03:32.195 ","End":"03:36.354","Text":"Then the second part, it\u0027s similar."},{"Start":"03:36.354 ","End":"03:39.510","Text":"This time when x is 2,"},{"Start":"03:39.510 ","End":"03:42.945","Text":"we get 0 because 2 minus x is 0."},{"Start":"03:42.945 ","End":"03:44.640","Text":"When x is 1,"},{"Start":"03:44.640 ","End":"03:46.150","Text":"then 2 minus 1 is 1."},{"Start":"03:46.150 ","End":"03:48.665","Text":"We basically get the same thing as here,"},{"Start":"03:48.665 ","End":"03:50.535","Text":"except with a minus,"},{"Start":"03:50.535 ","End":"03:52.610","Text":"so you see that\u0027s going to cancel out."},{"Start":"03:52.610 ","End":"03:54.230","Text":"Then the integral of this,"},{"Start":"03:54.230 ","End":"03:57.300","Text":"a minus with a minus give a plus."},{"Start":"03:57.310 ","End":"03:59.540","Text":"I had a plus here,"},{"Start":"03:59.540 ","End":"04:03.320","Text":"it should be a minus because we have minus minus minus."},{"Start":"04:03.320 ","End":"04:07.645","Text":"The integral of cosine is sine, no extra minus."},{"Start":"04:07.645 ","End":"04:11.200","Text":"We get the extra 2 over n Pi."},{"Start":"04:11.200 ","End":"04:13.385","Text":"Now, in the first one,"},{"Start":"04:13.385 ","End":"04:19.055","Text":"when x is 0, we get 0 because sine of 0 is 0."},{"Start":"04:19.055 ","End":"04:21.070","Text":"When we plug in 1,"},{"Start":"04:21.070 ","End":"04:25.080","Text":"what we get is we have a minus minus,"},{"Start":"04:25.080 ","End":"04:29.190","Text":"which is a plus 2^2 is 4 over n^2 Pi,"},{"Start":"04:29.190 ","End":"04:32.790","Text":"and the x is 1 sine n Pi over 2."},{"Start":"04:32.790 ","End":"04:35.850","Text":"Here also, when x is 2,"},{"Start":"04:35.850 ","End":"04:40.110","Text":"it\u0027s 0 because 2 and Pi over 2 gives n Pi,"},{"Start":"04:40.110 ","End":"04:41.610","Text":"so sine n Pi is 0,"},{"Start":"04:41.610 ","End":"04:43.520","Text":"and the 1 is subtracted,"},{"Start":"04:43.520 ","End":"04:45.830","Text":"which is what makes this minus a plus."},{"Start":"04:45.830 ","End":"04:47.630","Text":"And then we let x=1,"},{"Start":"04:47.630 ","End":"04:50.065","Text":"so we get n Pi over 2 here."},{"Start":"04:50.065 ","End":"04:53.390","Text":"Look what happens, this term will"},{"Start":"04:53.390 ","End":"04:56.660","Text":"cancel with this term because they have a minus and plus."},{"Start":"04:56.660 ","End":"04:58.430","Text":"But these two strengthen each other,"},{"Start":"04:58.430 ","End":"04:59.570","Text":"there is a plus and a plus,"},{"Start":"04:59.570 ","End":"05:02.335","Text":"so we\u0027ll get 8 times instead of 4."},{"Start":"05:02.335 ","End":"05:05.505","Text":"What we have is 8 over n^2 Pi squared,"},{"Start":"05:05.505 ","End":"05:08.705","Text":"sine n Pi over 2, and that\u0027s b_n."},{"Start":"05:08.705 ","End":"05:10.160","Text":"In a moment, we\u0027ll simplify it,"},{"Start":"05:10.160 ","End":"05:12.290","Text":"but this is the correct form of b_n."},{"Start":"05:12.290 ","End":"05:13.925","Text":"Now that we have b_n,"},{"Start":"05:13.925 ","End":"05:15.740","Text":"we know what f(x) is,"},{"Start":"05:15.740 ","End":"05:18.830","Text":"it\u0027s the sum of b_n sine n Pi x over 2."},{"Start":"05:18.830 ","End":"05:25.100","Text":"Just plug in the b_n from there and this is the answer for Fourier series of f,"},{"Start":"05:25.100 ","End":"05:32.690","Text":"except that I would like to simplify this sine n Pi over 2 into something more numerical."},{"Start":"05:32.690 ","End":"05:40.025","Text":"Now, sine n Pi over 2 is like sine of multiples of 90 degrees,"},{"Start":"05:40.025 ","End":"05:41.895","Text":"multiples of Pi over 2."},{"Start":"05:41.895 ","End":"05:43.425","Text":"We have Pi over 2,"},{"Start":"05:43.425 ","End":"05:45.690","Text":"Pi, 3 Pi over 2 Pi."},{"Start":"05:45.690 ","End":"05:48.060","Text":"It\u0027s like 90, 180,"},{"Start":"05:48.060 ","End":"05:50.735","Text":"270, 360, and then repeats."},{"Start":"05:50.735 ","End":"05:54.920","Text":"The sign of these will be going round the clock,"},{"Start":"05:54.920 ","End":"05:57.701","Text":"1, 0 minus 1,"},{"Start":"05:57.701 ","End":"06:01.010","Text":"0, 1, 0 minus 1, 0."},{"Start":"06:01.010 ","End":"06:04.070","Text":"In any event when n is even,"},{"Start":"06:04.070 ","End":"06:06.350","Text":"then n over 2 is a whole number,"},{"Start":"06:06.350 ","End":"06:08.390","Text":"so the sine is 0."},{"Start":"06:08.390 ","End":"06:11.540","Text":"So we only have to consider this when n is odd."},{"Start":"06:11.540 ","End":"06:14.450","Text":"When n is odd, let\u0027s say n is 2k minus 1,"},{"Start":"06:14.450 ","End":"06:17.855","Text":"it\u0027s your general odd number where k goes from 1 to infinity."},{"Start":"06:17.855 ","End":"06:22.525","Text":"So we have sine of n Pi over 2 is 2k minus 1 Pi over 2."},{"Start":"06:22.525 ","End":"06:24.215","Text":"With a bit of trigonometry,"},{"Start":"06:24.215 ","End":"06:27.875","Text":"this is equal to the sine of k Pi minus Pi over 2."},{"Start":"06:27.875 ","End":"06:31.910","Text":"I can switch the order and make it minus because sine is odd."},{"Start":"06:31.910 ","End":"06:35.900","Text":"There\u0027s a famous formula for the complimentary angle,"},{"Start":"06:35.900 ","End":"06:39.920","Text":"sine of Pi over 2 minus Alpha is cosine Alpha."},{"Start":"06:39.920 ","End":"06:43.670","Text":"So this just becomes cosine of k Pi."},{"Start":"06:43.670 ","End":"06:49.205","Text":"We know what cosine k Pi is, it\u0027s minus 1^k."},{"Start":"06:49.205 ","End":"06:55.445","Text":"We can throw the minus in and just increase the power of k and get it minus 1^k plus 1."},{"Start":"06:55.445 ","End":"06:57.380","Text":"Instead of the sine n Pi over 2,"},{"Start":"06:57.380 ","End":"06:58.705","Text":"we could put this."},{"Start":"06:58.705 ","End":"07:00.075","Text":"In the formula,"},{"Start":"07:00.075 ","End":"07:02.495","Text":"we get the sum of,"},{"Start":"07:02.495 ","End":"07:10.130","Text":"this is with the n. But if we replace n by 2k minus 1 and the sine with what we showed,"},{"Start":"07:10.130 ","End":"07:11.990","Text":"this becomes the sum."},{"Start":"07:11.990 ","End":"07:16.260","Text":"K goes from 1 to infinity and everything\u0027s in terms of 2k-1,"},{"Start":"07:16.260 ","End":"07:19.170","Text":"2k-1 squared Pi squared,"},{"Start":"07:19.170 ","End":"07:20.250","Text":"and then 8,"},{"Start":"07:20.250 ","End":"07:21.740","Text":"and then there\u0027s a sign here."},{"Start":"07:21.740 ","End":"07:23.555","Text":"Basically, instead of the sign,"},{"Start":"07:23.555 ","End":"07:25.650","Text":"I mean sign in trigonometric sense,"},{"Start":"07:25.650 ","End":"07:28.940","Text":"we have a sign in the plus-minus sense."},{"Start":"07:28.940 ","End":"07:30.740","Text":"It actually starts with a plus,"},{"Start":"07:30.740 ","End":"07:33.785","Text":"when k is 1, it starts with the plus, minus, plus, minus."},{"Start":"07:33.785 ","End":"07:39.420","Text":"Let\u0027s leave the answer as this and we are done."}],"ID":28797},{"Watched":false,"Name":"Exercise 6 - Part a","Duration":"5m 6s","ChapterTopicVideoID":27588,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.340","Text":"In this exercise, f is a real function which is periodic with"},{"Start":"00:05.340 ","End":"00:10.170","Text":"the period of 2 and on the interval from minus 1 to 1,"},{"Start":"00:10.170 ","End":"00:13.470","Text":"it\u0027s equal to 2 minus absolute value of x."},{"Start":"00:13.470 ","End":"00:15.180","Text":"This is what it looks like."},{"Start":"00:15.180 ","End":"00:18.270","Text":"This part here is 2 minus absolute value of x"},{"Start":"00:18.270 ","End":"00:21.285","Text":"and then we just copy paste, repeat infinitely,"},{"Start":"00:21.285 ","End":"00:24.245","Text":"we\u0027ll read each part as we come to it in Part a,"},{"Start":"00:24.245 ","End":"00:28.030","Text":"we want to expand f as a real Fourier series."},{"Start":"00:28.030 ","End":"00:31.220","Text":"Now this interval is not the standard minus Pi to Pi,"},{"Start":"00:31.220 ","End":"00:32.855","Text":"so it\u0027s a general interval."},{"Start":"00:32.855 ","End":"00:37.775","Text":"We\u0027ll use the formulas for a general interval which go from a to b."},{"Start":"00:37.775 ","End":"00:40.255","Text":"This is the formula."},{"Start":"00:40.255 ","End":"00:43.745","Text":"In our case, because f is an even function,"},{"Start":"00:43.745 ","End":"00:45.590","Text":"it\u0027ll just be a cosine series."},{"Start":"00:45.590 ","End":"00:46.670","Text":"There won\u0027t be any signs,"},{"Start":"00:46.670 ","End":"00:48.575","Text":"the bn will be 0."},{"Start":"00:48.575 ","End":"00:54.200","Text":"This is the formula for f and this is a naught and this is an,"},{"Start":"00:54.200 ","End":"00:58.835","Text":"now in our case, we know what a and b are minus 1 and 1."},{"Start":"00:58.835 ","End":"01:02.255","Text":"2 over b minus a is 1."},{"Start":"01:02.255 ","End":"01:05.250","Text":"But here is just Pi n x."},{"Start":"01:05.250 ","End":"01:06.915","Text":"If we plug all those in,"},{"Start":"01:06.915 ","End":"01:11.820","Text":"then we have that f(x) is like here,"},{"Start":"01:11.820 ","End":"01:14.505","Text":"but we replace this by Pi n x."},{"Start":"01:14.505 ","End":"01:18.120","Text":"A naught is the integral from minus 1 to 1,"},{"Start":"01:18.120 ","End":"01:21.265","Text":"and this is just 1, and an is this."},{"Start":"01:21.265 ","End":"01:24.815","Text":"Because f is an even function,"},{"Start":"01:24.815 ","End":"01:27.740","Text":"also f times cosine is an even function."},{"Start":"01:27.740 ","End":"01:31.970","Text":"We can just take the integrals from 0 to 1 and double them."},{"Start":"01:31.970 ","End":"01:33.980","Text":"Now these are the formulas for a naught,"},{"Start":"01:33.980 ","End":"01:35.785","Text":"and an let\u0027s compute them."},{"Start":"01:35.785 ","End":"01:37.700","Text":"First, with a naught."},{"Start":"01:37.700 ","End":"01:43.585","Text":"A naught is twice the integral from 0-1 of 2 minus x dx."},{"Start":"01:43.585 ","End":"01:48.440","Text":"A straightforward integral, integral of 2 minus x into x minus x^2 over 2."},{"Start":"01:48.440 ","End":"01:51.036","Text":"If you plug in 0, you get nothing, if you plug in 1,"},{"Start":"01:51.036 ","End":"01:54.765","Text":"well, you get the answer of 3."},{"Start":"01:54.765 ","End":"01:56.445","Text":"That\u0027s a naught."},{"Start":"01:56.445 ","End":"01:58.855","Text":"A n is this,"},{"Start":"01:58.855 ","End":"02:05.340","Text":"just what we have written here with f(x) equaling 2 minus x."},{"Start":"02:05.410 ","End":"02:08.900","Text":"We\u0027ll do this using integration by parts."},{"Start":"02:08.900 ","End":"02:11.429","Text":"I won\u0027t give the formula this time,"},{"Start":"02:11.429 ","End":"02:13.670","Text":"you should really know it already."},{"Start":"02:13.670 ","End":"02:15.860","Text":"Basically decide one of these you\u0027re going"},{"Start":"02:15.860 ","End":"02:18.245","Text":"differentiate and one of these you\u0027re going to integrate."},{"Start":"02:18.245 ","End":"02:22.370","Text":"What we\u0027ll do is, let\u0027s say this is fg\u0027, so this is wrong."},{"Start":"02:22.370 ","End":"02:25.540","Text":"We need to differentiate and this, to integrate."},{"Start":"02:25.540 ","End":"02:27.515","Text":"First of all, f, g,"},{"Start":"02:27.515 ","End":"02:29.600","Text":"g would be the integral of this."},{"Start":"02:29.600 ","End":"02:32.150","Text":"It\u0027s like sine of n Pi x,"},{"Start":"02:32.150 ","End":"02:38.555","Text":"but divided by n Pi in a derivative minus the integral."},{"Start":"02:38.555 ","End":"02:42.785","Text":"This time this is differentiated so minus 1."},{"Start":"02:42.785 ","End":"02:47.195","Text":"This bit, again, here."},{"Start":"02:47.195 ","End":"02:51.035","Text":"This bit is equal to 0."},{"Start":"02:51.035 ","End":"02:53.975","Text":"Plug in x=0,"},{"Start":"02:53.975 ","End":"02:55.445","Text":"we get sine of 0,"},{"Start":"02:55.445 ","End":"02:57.770","Text":"if you plug in x=1,"},{"Start":"02:57.770 ","End":"03:01.870","Text":"we get sine of n Pi is also 0."},{"Start":"03:01.870 ","End":"03:04.245","Text":"We\u0027re just left with this bit,"},{"Start":"03:04.245 ","End":"03:08.195","Text":"which is twice minus and the minus cancel."},{"Start":"03:08.195 ","End":"03:10.925","Text":"The n Pi comes out in front."},{"Start":"03:10.925 ","End":"03:14.360","Text":"The integral of sine of n Pi x dx."},{"Start":"03:14.360 ","End":"03:17.660","Text":"Integral of sine is minus cosine."},{"Start":"03:17.660 ","End":"03:21.590","Text":"But we also have to divide by another n Pi,"},{"Start":"03:21.590 ","End":"03:23.945","Text":"making it n^2 Pi^2."},{"Start":"03:23.945 ","End":"03:26.315","Text":"Then from 0-1."},{"Start":"03:26.315 ","End":"03:29.735","Text":"If we let x=1,"},{"Start":"03:29.735 ","End":"03:32.090","Text":"we have cosine of n Pi,"},{"Start":"03:32.090 ","End":"03:37.500","Text":"which is minus 1^n and cosine of 0 is 1."},{"Start":"03:37.570 ","End":"03:42.455","Text":"Now this depends on whether n is odd or even."},{"Start":"03:42.455 ","End":"03:45.680","Text":"Minus 1 to the n will either be 1 or minus 1."},{"Start":"03:45.680 ","End":"03:47.480","Text":"Let\u0027s separate."},{"Start":"03:47.480 ","End":"03:49.370","Text":"If n is odd,"},{"Start":"03:49.370 ","End":"03:52.595","Text":"2k minus 1,then we have,"},{"Start":"03:52.595 ","End":"03:59.085","Text":"this is minus 1 minus 1 is minus 2 times minus 2 is 4."},{"Start":"03:59.085 ","End":"04:05.210","Text":"With an even, it\u0027s just minus 1 plus 1 is 0. We\u0027ve got this."},{"Start":"04:05.210 ","End":"04:14.135","Text":"Then we can substitute this in the formula that we had for the sum of a n cosine n Pi x,"},{"Start":"04:14.135 ","End":"04:17.685","Text":"that n becomes 2k minus 1."},{"Start":"04:17.685 ","End":"04:19.680","Text":"A naught was 3,"},{"Start":"04:19.680 ","End":"04:23.130","Text":"so a naught over 2 is 3 over 2."},{"Start":"04:23.130 ","End":"04:25.054","Text":"This is the answer."},{"Start":"04:25.054 ","End":"04:29.840","Text":"I\u0027ll show you another way of doing it based on the previous exercise we did."},{"Start":"04:29.840 ","End":"04:32.180","Text":"In the previous exercise,"},{"Start":"04:32.180 ","End":"04:35.030","Text":"we showed that the absolute value of x on"},{"Start":"04:35.030 ","End":"04:40.045","Text":"the same interval from minus 1 to 1 as the Fourier expansion, like so."},{"Start":"04:40.045 ","End":"04:43.820","Text":"If we want 2 minus absolute value of x,"},{"Start":"04:43.820 ","End":"04:49.360","Text":"just subtract both sides from 2 and we get 2 minus absolute value of x."},{"Start":"04:49.360 ","End":"04:53.125","Text":"Here we get 2 minus a 1/2 is 3 over 2,"},{"Start":"04:53.125 ","End":"04:54.485","Text":"and the minus width,"},{"Start":"04:54.485 ","End":"04:56.405","Text":"this makes it a plus."},{"Start":"04:56.405 ","End":"04:59.135","Text":"If you subtract this from 2, you get this."},{"Start":"04:59.135 ","End":"05:03.620","Text":"If we\u0027d had that previous exercise and we can just do it in one line."},{"Start":"05:03.620 ","End":"05:06.210","Text":"That concludes Part a."}],"ID":28798},{"Watched":false,"Name":"Exercise 6 - Part b","Duration":"2m 49s","ChapterTopicVideoID":27589,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.255","Text":"Now, we come to part b,"},{"Start":"00:03.255 ","End":"00:06.300","Text":"where we have to compute the sum of this series."},{"Start":"00:06.300 ","End":"00:08.475","Text":"Remember, in part a,"},{"Start":"00:08.475 ","End":"00:14.144","Text":"we had the Fourier expansion of 2 minus absolute value of x, was this."},{"Start":"00:14.144 ","End":"00:16.950","Text":"There\u0027s 1 of 2 tools we could use."},{"Start":"00:16.950 ","End":"00:23.940","Text":"It\u0027s usually either the Dirichlet\u0027s theorem or it\u0027s the Parseval identity."},{"Start":"00:23.940 ","End":"00:26.910","Text":"Because here it\u0027s squared and here it\u0027s to the fourth,"},{"Start":"00:26.910 ","End":"00:29.295","Text":"we\u0027re going to go with Parseval."},{"Start":"00:29.295 ","End":"00:34.610","Text":"Parseval\u0027s identity for a general integral from a to b is this."},{"Start":"00:34.610 ","End":"00:40.295","Text":"I faded out the b_n because we know the b_n is a 0 in our case."},{"Start":"00:40.295 ","End":"00:48.080","Text":"What we get, since we know that a is minus 1 and b is 1 and so on we can simplify this."},{"Start":"00:48.080 ","End":"00:51.950","Text":"Adapted to our case and 2 over b minus a is 1,"},{"Start":"00:51.950 ","End":"00:54.920","Text":"so it\u0027s just absolute value of f(x)^2 dx."},{"Start":"00:54.920 ","End":"00:58.730","Text":"I meant to say we\u0027re just going to work on the left-hand side first."},{"Start":"00:58.730 ","End":"01:05.045","Text":"This is twice the integral from 0-1 because it\u0027s an even function."},{"Start":"01:05.045 ","End":"01:12.150","Text":"Absolute value of f(x) is absolute value of 2 minus x, or 2 minus absolute value of x."},{"Start":"01:12.610 ","End":"01:15.215","Text":"Anyway, when x is positive,"},{"Start":"01:15.215 ","End":"01:18.050","Text":"this is just (2-x)^2."},{"Start":"01:18.050 ","End":"01:20.840","Text":"Let\u0027s see this integral."},{"Start":"01:20.840 ","End":"01:24.470","Text":"We can raise the power by 1, make it 3,"},{"Start":"01:24.470 ","End":"01:30.115","Text":"divide by 3 but we also have to divide by minus 1."},{"Start":"01:30.115 ","End":"01:32.150","Text":"Let\u0027s see the inner derivative."},{"Start":"01:32.150 ","End":"01:35.480","Text":"This is what we get and we have to evaluate between 0 and 1."},{"Start":"01:35.480 ","End":"01:43.890","Text":"At 0 we get (2-0)^3 here and (2-1)^3."},{"Start":"01:43.890 ","End":"01:45.600","Text":"This comes out to be 8."},{"Start":"01:45.600 ","End":"01:51.709","Text":"This is 1, so it\u0027s 7 times 2/3 is 14/3."},{"Start":"01:51.709 ","End":"01:55.910","Text":"The 14/3 is the left-hand side that goes here."},{"Start":"01:55.910 ","End":"01:58.460","Text":"That\u0027s the integral equals."},{"Start":"01:58.460 ","End":"02:02.655","Text":"Now, a_0 was 3,"},{"Start":"02:02.655 ","End":"02:09.550","Text":"and a_n we found was equal to this."},{"Start":"02:09.560 ","End":"02:12.480","Text":"Now, some fractions,"},{"Start":"02:12.480 ","End":"02:16.920","Text":"14/3 minus 9/2 is what?"},{"Start":"02:16.920 ","End":"02:21.360","Text":"28 minus 27 over 6, 1/6."},{"Start":"02:21.360 ","End":"02:26.825","Text":"Here we can take 4^2 over Pi squared squared outside."},{"Start":"02:26.825 ","End":"02:33.845","Text":"We don\u0027t need the absolute value because it\u0027s really squared and 1/(2k-1)^2, the sum of."},{"Start":"02:33.845 ","End":"02:39.785","Text":"All we have to do now is switch sides and bring the 16/Pi^4 to the other side."},{"Start":"02:39.785 ","End":"02:46.255","Text":"We get Pi^4/6x16 will be Pi^4/96."},{"Start":"02:46.255 ","End":"02:50.620","Text":"That\u0027s the answer and this is part b."}],"ID":28799},{"Watched":false,"Name":"Exercise 6- Part c","Duration":"1m 36s","ChapterTopicVideoID":27590,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.639","Text":"Continuing with this exercise,"},{"Start":"00:02.639 ","End":"00:05.850","Text":"we now come to part C. Again,"},{"Start":"00:05.850 ","End":"00:07.770","Text":"we have some of the series."},{"Start":"00:07.770 ","End":"00:09.195","Text":"In part B,"},{"Start":"00:09.195 ","End":"00:11.640","Text":"we used possibles identity,"},{"Start":"00:11.640 ","End":"00:15.225","Text":"in part C, we\u0027ll use Dirichlet\u0027s theorem."},{"Start":"00:15.225 ","End":"00:20.820","Text":"Now in part A, we found that this is the Fourier series that represents"},{"Start":"00:20.820 ","End":"00:26.085","Text":"2 minus absolute value of x on the interval minus 1 to 1."},{"Start":"00:26.085 ","End":"00:30.345","Text":"Notice that at x=0,"},{"Start":"00:30.345 ","End":"00:32.820","Text":"we have a continuous function."},{"Start":"00:32.820 ","End":"00:35.070","Text":"The function is continuous at x= 0."},{"Start":"00:35.070 ","End":"00:36.885","Text":"In fact, it\u0027s continuous everywhere."},{"Start":"00:36.885 ","End":"00:40.065","Text":"Because of that, we can apply"},{"Start":"00:40.065 ","End":"00:47.670","Text":"Dirichlet\u0027s theorem and substitute x=0 and put equality instead of this tilde."},{"Start":"00:47.670 ","End":"00:54.285","Text":"2 minus absolute value of 0 is equal to all this and there\u0027s a 0 here."},{"Start":"00:54.285 ","End":"00:57.780","Text":"This makes this cosine 0, which is 1."},{"Start":"00:57.780 ","End":"01:01.010","Text":"All of this part can be removed and so can the"},{"Start":"01:01.010 ","End":"01:05.090","Text":"minus 0 and we can bring the 3 over 2 over to"},{"Start":"01:05.090 ","End":"01:08.950","Text":"the left side so we get 2 minus 3 over 2 is a half"},{"Start":"01:08.950 ","End":"01:15.075","Text":"equals and here take the 4 over Pi^2 in front of the summation."},{"Start":"01:15.075 ","End":"01:19.640","Text":"We get this. Now all we have to do is swap sides and"},{"Start":"01:19.640 ","End":"01:25.560","Text":"multiply by the reciprocal of this Pi^2 over 4."},{"Start":"01:25.560 ","End":"01:29.250","Text":"This sum becomes Pi^2 over 4 times a half,"},{"Start":"01:29.250 ","End":"01:31.530","Text":"which is Pi^2 over 8,"},{"Start":"01:31.530 ","End":"01:36.340","Text":"and that\u0027s the answer to part C."}],"ID":28800},{"Watched":false,"Name":"Exercise 6 - Part d","Duration":"2m 19s","ChapterTopicVideoID":27575,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.970","Text":"Continuing the exercise, we come to the last part which is D, and the question is,"},{"Start":"00:05.970 ","End":"00:08.320","Text":"does the Fourier series for f,"},{"Start":"00:08.320 ","End":"00:10.260","Text":"that we had in part A,"},{"Start":"00:10.260 ","End":"00:15.630","Text":"does it converge uniformly on the interval minus 1,1?"},{"Start":"00:15.630 ","End":"00:18.435","Text":"The answer it turns out is yes."},{"Start":"00:18.435 ","End":"00:21.160","Text":"We\u0027ll apply the following proposition."},{"Start":"00:21.160 ","End":"00:25.352","Text":"Here it is in the context in which it was originally shown."},{"Start":"00:25.352 ","End":"00:29.165","Text":"This is paraphrased, just slightly abbreviated."},{"Start":"00:29.165 ","End":"00:34.610","Text":"Basically we have f and f\u0027 piecewise continuous on an interval a,"},{"Start":"00:34.610 ","End":"00:39.530","Text":"b and the function f itself has to be continuous,"},{"Start":"00:39.530 ","End":"00:43.760","Text":"not just piecewise and has to agree at the endpoints."},{"Start":"00:43.760 ","End":"00:49.010","Text":"Then the Fourier series for f converges uniformly to f on the interval."},{"Start":"00:49.010 ","End":"00:52.250","Text":"I claim that our f matches these conditions."},{"Start":"00:52.250 ","End":"00:54.770","Text":"F is clearly continuous."},{"Start":"00:54.770 ","End":"01:00.870","Text":"I mean, even by the formula 2 minus absolute value of x certainly continuous,"},{"Start":"01:00.870 ","End":"01:05.390","Text":"and we mentioned that also in part C. Let\u0027s see the next condition,"},{"Start":"01:05.390 ","End":"01:08.345","Text":"f\u0027 is piecewise continuous."},{"Start":"01:08.345 ","End":"01:14.520","Text":"Let\u0027s see; f\u0027 is a step function,"},{"Start":"01:14.520 ","End":"01:17.535","Text":"it\u0027s like minus 1 here,"},{"Start":"01:17.535 ","End":"01:20.695","Text":"1 here, minus 1, then 1."},{"Start":"01:20.695 ","End":"01:23.810","Text":"The only place it doesn\u0027t exist or it\u0027s not"},{"Start":"01:23.810 ","End":"01:27.155","Text":"continuous are these points which are integers,"},{"Start":"01:27.155 ","End":"01:29.550","Text":"where it jumps from 1 to minus 1 here,"},{"Start":"01:29.550 ","End":"01:32.580","Text":"from minus 1 to 1 to minus 1."},{"Start":"01:32.580 ","End":"01:34.110","Text":"Except at these points,"},{"Start":"01:34.110 ","End":"01:35.805","Text":"it is continuous,"},{"Start":"01:35.805 ","End":"01:38.805","Text":"so it is piecewise continuous."},{"Start":"01:38.805 ","End":"01:41.235","Text":"As for this condition,"},{"Start":"01:41.235 ","End":"01:44.735","Text":"f is periodic with a period of 2,"},{"Start":"01:44.735 ","End":"01:49.265","Text":"so if you let x equals minus 1 and f of minus 1 equals f of 1,"},{"Start":"01:49.265 ","End":"01:54.280","Text":"or you could just check it directly by plugging in minus 1 or 1"},{"Start":"01:54.280 ","End":"02:00.020","Text":"to 2 minus absolute value of x and in both cases you get the same answer, which is 1."},{"Start":"02:00.020 ","End":"02:04.420","Text":"F meets the condition of this proposition,"},{"Start":"02:04.420 ","End":"02:10.370","Text":"and so we get the conclusion that the Fourier series converges uniformly."},{"Start":"02:10.370 ","End":"02:13.460","Text":"More than that it converges uniformly to the original f,"},{"Start":"02:13.460 ","End":"02:15.680","Text":"2 minus absolute value of x."},{"Start":"02:15.680 ","End":"02:20.010","Text":"That concludes part D and this exercise."}],"ID":28801},{"Watched":false,"Name":"Sine and Cosine Series of a Function","Duration":"5m 56s","ChapterTopicVideoID":27576,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.335","Text":"In this clip, we\u0027ll talk about sine and cosine series of a function."},{"Start":"00:04.335 ","End":"00:07.560","Text":"These are special kinds of Fourier series."},{"Start":"00:07.560 ","End":"00:13.950","Text":"We start with a piecewise continuous function f on an interval from 0"},{"Start":"00:13.950 ","End":"00:20.370","Text":"to L. We\u0027ve seen before that we can expand f as a sine series or a cosine series."},{"Start":"00:20.370 ","End":"00:23.475","Text":"But there are some shortcut formulas."},{"Start":"00:23.475 ","End":"00:30.533","Text":"What we usually did was extended f to a symmetric interval and made it odd or even,"},{"Start":"00:30.533 ","End":"00:33.179","Text":"and then we got just signs are just cosines,"},{"Start":"00:33.179 ","End":"00:36.850","Text":"and then we restricted the answer back to 0,"},{"Start":"00:36.850 ","End":"00:40.070","Text":"L. We can save some computation,"},{"Start":"00:40.070 ","End":"00:44.615","Text":"go straight to the formula for cosine series and sine series in a moment."},{"Start":"00:44.615 ","End":"00:50.720","Text":"The cosine series is the following on a general interval 0,"},{"Start":"00:50.720 ","End":"00:54.779","Text":"L and this is what the formula is."},{"Start":"00:54.779 ","End":"00:59.915","Text":"We get the coefficients a_n by the following formula."},{"Start":"00:59.915 ","End":"01:02.284","Text":"The sine series very similar."},{"Start":"01:02.284 ","End":"01:04.370","Text":"Instead of cosine, we have sine,"},{"Start":"01:04.370 ","End":"01:07.349","Text":"we don\u0027t need this a_Naught over 2 here,"},{"Start":"01:07.349 ","End":"01:11.315","Text":"and this is the formula for the coefficient b_n."},{"Start":"01:11.315 ","End":"01:13.370","Text":"That\u0027s all there is to it."},{"Start":"01:13.370 ","End":"01:16.300","Text":"Let\u0027s do a couple of examples 1 of each."},{"Start":"01:16.300 ","End":"01:23.075","Text":"First example, find the cosine series for f(x) equals x on the interval 0 Pi."},{"Start":"01:23.075 ","End":"01:28.750","Text":"Now here\u0027s the formula for cosine series."},{"Start":"01:28.750 ","End":"01:33.330","Text":"In our case, L is equal to Pi."},{"Start":"01:33.330 ","End":"01:35.535","Text":"What we get is, well,"},{"Start":"01:35.535 ","End":"01:38.910","Text":"Pi over L cancels and we just have nx here."},{"Start":"01:38.910 ","End":"01:45.170","Text":"Here, we replace L by Pi here and here and here also Pi over L cancels,"},{"Start":"01:45.170 ","End":"01:51.770","Text":"so it just nx and f(x) of course is equal to x on 0 Pi."},{"Start":"01:51.770 ","End":"01:56.510","Text":"We have to compute this integral and will do it by parts."},{"Start":"01:56.510 ","End":"01:58.795","Text":"I won\u0027t repeat the formula."},{"Start":"01:58.795 ","End":"02:03.120","Text":"Let\u0027s say this is f and this is g\u0027."},{"Start":"02:03.120 ","End":"02:06.550","Text":"Here\u0027s f, here\u0027s g, which is cosine,"},{"Start":"02:06.550 ","End":"02:13.105","Text":"which is sine nx over n. Here\u0027s f\u0027 and here\u0027s g. Again,"},{"Start":"02:13.105 ","End":"02:15.100","Text":"this won\u0027t work if n equals 0."},{"Start":"02:15.100 ","End":"02:17.995","Text":"We\u0027ll have to compute a 0 separately."},{"Start":"02:17.995 ","End":"02:23.350","Text":"Now, this is 0 because sine of 0 is 0,"},{"Start":"02:23.350 ","End":"02:25.690","Text":"and sine of n Pi is 0."},{"Start":"02:25.690 ","End":"02:29.485","Text":"This part drops off so we just have this integral."},{"Start":"02:29.485 ","End":"02:33.545","Text":"When we integrate minus sine, we get cosine,"},{"Start":"02:33.545 ","End":"02:41.105","Text":"but we have to divide by another n so that makes it n squared between 0 and Pi."},{"Start":"02:41.105 ","End":"02:46.400","Text":"We get cosine n Pi minus cosine n 0 over n squared."},{"Start":"02:46.400 ","End":"02:52.350","Text":"Now cosine n Pi is minus 1 to the n. Cosine 0 is 1."},{"Start":"02:52.350 ","End":"02:54.435","Text":"This is what we have."},{"Start":"02:54.435 ","End":"02:58.400","Text":"We\u0027re going to separate odds and evens because minus 1 to the n will"},{"Start":"02:58.400 ","End":"03:02.525","Text":"either be 1 or minus 1 accordingly."},{"Start":"03:02.525 ","End":"03:04.900","Text":"If n is odd,"},{"Start":"03:04.900 ","End":"03:07.260","Text":"say 2k minus 1,"},{"Start":"03:07.260 ","End":"03:13.755","Text":"then we get minus 1 minus 1 is minus 2 times 2 is minus over Pi n^2."},{"Start":"03:13.755 ","End":"03:17.565","Text":"But we replace n by 2k minus 1."},{"Start":"03:17.565 ","End":"03:20.070","Text":"If n is even,"},{"Start":"03:20.070 ","End":"03:22.260","Text":"then we have 1 minus 1 is 0."},{"Start":"03:22.260 ","End":"03:23.730","Text":"This is all 0."},{"Start":"03:23.730 ","End":"03:27.575","Text":"Whoever we still haven\u0027t computed a 0, a_Naught,"},{"Start":"03:27.575 ","End":"03:32.735","Text":"which is just the integral of f(x) dx 2 over Pi."},{"Start":"03:32.735 ","End":"03:38.740","Text":"That comes out to be Pi in the n is a straightforward integral."},{"Start":"03:38.740 ","End":"03:41.915","Text":"Now we have a_n and we have a_Naught."},{"Start":"03:41.915 ","End":"03:47.675","Text":"Formula from above is this f(x) in terms of a_Naught."},{"Start":"03:47.675 ","End":"03:49.340","Text":"We have a_Naught in a_n,"},{"Start":"03:49.340 ","End":"03:51.205","Text":"so just plug them in."},{"Start":"03:51.205 ","End":"03:52.920","Text":"When we plug in the a_n,"},{"Start":"03:52.920 ","End":"03:57.460","Text":"we\u0027re going to replace n by k. A_Naught is Pi over 2,"},{"Start":"03:57.460 ","End":"04:01.685","Text":"and then we have the sum of this coefficient."},{"Start":"04:01.685 ","End":"04:06.055","Text":"Cosine nx is cosine 2k minus 1x."},{"Start":"04:06.055 ","End":"04:08.330","Text":"That\u0027s the answer."},{"Start":"04:08.330 ","End":"04:14.315","Text":"Next example, this time a sine series f(x) equals 1 on 0, Pi."},{"Start":"04:14.315 ","End":"04:18.925","Text":"Again, the general formulas of these,"},{"Start":"04:18.925 ","End":"04:22.245","Text":"and in our case, L equals Pi."},{"Start":"04:22.245 ","End":"04:27.005","Text":"We get Pi over L cancels,"},{"Start":"04:27.005 ","End":"04:31.060","Text":"and here we replace L by Pi."},{"Start":"04:31.060 ","End":"04:33.060","Text":"Also, f(x) is 1,"},{"Start":"04:33.060 ","End":"04:34.620","Text":"we need to compute b_n."},{"Start":"04:34.620 ","End":"04:38.060","Text":"B_n from 1,2,3 to infinity."},{"Start":"04:38.060 ","End":"04:40.280","Text":"B_n is this integral,"},{"Start":"04:40.280 ","End":"04:42.440","Text":"and that\u0027s a straightforward integral."},{"Start":"04:42.440 ","End":"04:45.200","Text":"The integral of sine is minus cosine."},{"Start":"04:45.200 ","End":"04:52.475","Text":"But instead of the minus switch the limits of integration, so it compensates."},{"Start":"04:52.475 ","End":"04:54.650","Text":"Here, n is bigger than 0."},{"Start":"04:54.650 ","End":"04:58.147","Text":"We don\u0027t need to worry about a special case where n is 0,"},{"Start":"04:58.147 ","End":"05:00.620","Text":"n won\u0027t be 0 for the b_n."},{"Start":"05:00.620 ","End":"05:06.880","Text":"This is equal to cosine 0 minus cosine n Pi over n. Once again,"},{"Start":"05:06.880 ","End":"05:11.100","Text":"cosine nPi is minus 1 to the n. Again,"},{"Start":"05:11.100 ","End":"05:13.980","Text":"we split odds and evens."},{"Start":"05:13.980 ","End":"05:16.740","Text":"If n is odd,"},{"Start":"05:16.740 ","End":"05:18.300","Text":"we get 1 minus,"},{"Start":"05:18.300 ","End":"05:21.195","Text":"minus 1 is 2 times 2 is 4,"},{"Start":"05:21.195 ","End":"05:24.650","Text":"n is 2k minus 1, and Pi here."},{"Start":"05:24.650 ","End":"05:27.275","Text":"That\u0027s a general odd number."},{"Start":"05:27.275 ","End":"05:28.610","Text":"When n is even,"},{"Start":"05:28.610 ","End":"05:30.305","Text":"it\u0027s 1 minus 1 is 0,"},{"Start":"05:30.305 ","End":"05:33.365","Text":"it\u0027s just 0 when n is even or n equals 2k."},{"Start":"05:33.365 ","End":"05:35.555","Text":"Now that we have b_n,"},{"Start":"05:35.555 ","End":"05:40.490","Text":"we can plug that into the formula where f(x) is the sum of b_n sine and x."},{"Start":"05:40.490 ","End":"05:42.925","Text":"Now we have b_n, which is this."},{"Start":"05:42.925 ","End":"05:45.075","Text":"We changed the indexing to k,"},{"Start":"05:45.075 ","End":"05:47.355","Text":"k from 1 to infinity."},{"Start":"05:47.355 ","End":"05:49.990","Text":"I think I wrote minus 4, it\u0027s just 4,"},{"Start":"05:49.990 ","End":"05:53.055","Text":"4 over 2k minus 1 Pi sine 2k minus 1x."},{"Start":"05:53.055 ","End":"05:57.010","Text":"That\u0027s the answer. That concludes this clip."}],"ID":28802},{"Watched":false,"Name":"Exercise 7","Duration":"3m 26s","ChapterTopicVideoID":27577,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.290","Text":"In this exercise, we\u0027re going to find the cosine series for f(x)=x on the interval 0, 3."},{"Start":"00:07.290 ","End":"00:11.745","Text":"The formula for the cosine series is this."},{"Start":"00:11.745 ","End":"00:13.515","Text":"We want to find a_n."},{"Start":"00:13.515 ","End":"00:16.020","Text":"Well, we know that l is 3."},{"Start":"00:16.020 ","End":"00:18.930","Text":"Just replace it here, here and here."},{"Start":"00:18.930 ","End":"00:24.900","Text":"We get 2/3 interval from 0-3 x cosine n Pi x over 3 dx,"},{"Start":"00:24.900 ","End":"00:28.730","Text":"and we\u0027ll do this as an integration by parts."},{"Start":"00:28.730 ","End":"00:34.500","Text":"The x will be u and the cosine n Pi x over 3 will be v\u0027."},{"Start":"00:34.500 ","End":"00:39.105","Text":"We need to find v and u\u0027."},{"Start":"00:39.105 ","End":"00:41.505","Text":"U\u0027 is just 1."},{"Start":"00:41.505 ","End":"00:49.510","Text":"V is sine of n Pi x over 3,"},{"Start":"00:49.510 ","End":"00:56.105","Text":"but we have to divide by the constant times x. I have to divide by it."},{"Start":"00:56.105 ","End":"01:00.824","Text":"It\u0027s 3 over n Pi with the x and then in the other integral"},{"Start":"01:00.824 ","End":"01:06.030","Text":"1 which is the u\u0027 and v the same as here,"},{"Start":"01:06.030 ","End":"01:07.575","Text":"sine n Pi x over 3,"},{"Start":"01:07.575 ","End":"01:09.700","Text":"3 over n Pi."},{"Start":"01:09.700 ","End":"01:14.450","Text":"This part, which I\u0027ve done in gray is 0."},{"Start":"01:14.450 ","End":"01:17.815","Text":"Because if you plug in 0, it\u0027s 0."},{"Start":"01:17.815 ","End":"01:19.590","Text":"If plug in 3,"},{"Start":"01:19.590 ","End":"01:23.450","Text":"n Pi 3 over 3 is n Pi sine of n Pi is 0."},{"Start":"01:23.450 ","End":"01:25.490","Text":"We just have this part."},{"Start":"01:25.490 ","End":"01:27.575","Text":"The integral of this,"},{"Start":"01:27.575 ","End":"01:31.610","Text":"is integral of minus sine is cosine."},{"Start":"01:31.610 ","End":"01:38.300","Text":"But we have to again divide by n Pi over 3 or multiply by 3 over n Pi,"},{"Start":"01:38.300 ","End":"01:42.145","Text":"so becomes squared and the 2/3 is from here."},{"Start":"01:42.145 ","End":"01:44.470","Text":"Now if we plug in 3,"},{"Start":"01:44.470 ","End":"01:47.840","Text":"this part becomes cosine n Pi, plugging in 0."},{"Start":"01:47.840 ","End":"01:49.715","Text":"This part is 1."},{"Start":"01:49.715 ","End":"01:56.885","Text":"We have this constant which comes out to be 6 over n^2 Pi^2 and then the difference"},{"Start":"01:56.885 ","End":"01:59.450","Text":"of cosine n Pi minus 1,"},{"Start":"01:59.450 ","End":"02:03.680","Text":"we know that cosine n Pi is minus 1 to the n. We\u0027ve seen this many times."},{"Start":"02:03.680 ","End":"02:06.995","Text":"We split into n odd or n even."},{"Start":"02:06.995 ","End":"02:08.450","Text":"We do in cases like this,"},{"Start":"02:08.450 ","End":"02:10.985","Text":"so if n is odd,"},{"Start":"02:10.985 ","End":"02:14.245","Text":"typical odd number is 2k minus 1,"},{"Start":"02:14.245 ","End":"02:17.280","Text":"then we have here minus 1,"},{"Start":"02:17.280 ","End":"02:19.070","Text":"minus 1 is minus 2,"},{"Start":"02:19.070 ","End":"02:21.320","Text":"so it\u0027s minus 12 here."},{"Start":"02:21.320 ","End":"02:24.470","Text":"Instead of n, we put 2k minus 1."},{"Start":"02:24.470 ","End":"02:26.630","Text":"If n is even to k,"},{"Start":"02:26.630 ","End":"02:29.860","Text":"then it\u0027s 0, I should\u0027ve mentioned."},{"Start":"02:29.860 ","End":"02:34.730","Text":"This development from this line to this line doesn\u0027t work if n is 0,"},{"Start":"02:34.730 ","End":"02:37.115","Text":"because we\u0027re dividing by n,"},{"Start":"02:37.115 ","End":"02:40.430","Text":"will have to compute a_0 separately,"},{"Start":"02:40.430 ","End":"02:42.440","Text":"which is often the case."},{"Start":"02:42.440 ","End":"02:49.380","Text":"A_0 is 2/3 interval of 0-3 f(x) xdx."},{"Start":"02:49.380 ","End":"02:52.425","Text":"This is a very simple integral."},{"Start":"02:52.425 ","End":"02:54.170","Text":"I\u0027ll leave you to look at it."},{"Start":"02:54.170 ","End":"02:56.255","Text":"The answer is a_0 =3."},{"Start":"02:56.255 ","End":"02:58.955","Text":"Here\u0027s a_n and here\u0027s a_0."},{"Start":"02:58.955 ","End":"03:02.180","Text":"Now f(x), this was the formula,"},{"Start":"03:02.180 ","End":"03:06.640","Text":"just will put a_0 and a_n from here and here,"},{"Start":"03:06.640 ","End":"03:09.855","Text":"and l is 3."},{"Start":"03:09.855 ","End":"03:14.790","Text":"We get 3 over 2 plus and then instead of n,"},{"Start":"03:14.790 ","End":"03:16.905","Text":"we put 2k minus 1."},{"Start":"03:16.905 ","End":"03:18.210","Text":"This should be a 3."},{"Start":"03:18.210 ","End":"03:19.885","Text":"We know what l is."},{"Start":"03:19.885 ","End":"03:21.695","Text":"This is the answer."},{"Start":"03:21.695 ","End":"03:26.310","Text":"There is no more to be done and that concludes this exercise."}],"ID":28803},{"Watched":false,"Name":"Exercise 8","Duration":"5m 51s","ChapterTopicVideoID":27578,"CourseChapterTopicPlaylistID":294458,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"In this exercise, we have to find"},{"Start":"00:03.660 ","End":"00:10.409","Text":"the sine series for f(x)=cosine 2x on the interval 0 to Pi."},{"Start":"00:10.409 ","End":"00:13.920","Text":"There\u0027s a formula for the sine series of a function f,"},{"Start":"00:13.920 ","End":"00:15.629","Text":"a pair of formulas."},{"Start":"00:15.629 ","End":"00:20.880","Text":"f(x) is represented by sum of b_n sine nPix over"},{"Start":"00:20.880 ","End":"00:26.265","Text":"L and b_n is given by the following formula."},{"Start":"00:26.265 ","End":"00:28.590","Text":"In our case, L=Pi."},{"Start":"00:28.590 ","End":"00:32.175","Text":"So if we plug that in, we\u0027ll get this."},{"Start":"00:32.175 ","End":"00:35.550","Text":"The other formula adapts as follows."},{"Start":"00:35.550 ","End":"00:39.088","Text":"So b_n is this integral."},{"Start":"00:39.088 ","End":"00:41.480","Text":"Replace f(x) by what we know it is,"},{"Start":"00:41.480 ","End":"00:43.145","Text":"which is cosine 2x."},{"Start":"00:43.145 ","End":"00:46.190","Text":"This is equal to using"},{"Start":"00:46.190 ","End":"00:50.974","Text":"the trigonometric identity for cosine of something, sine of something."},{"Start":"00:50.974 ","End":"00:52.970","Text":"This is the formula here,"},{"Start":"00:52.970 ","End":"00:56.465","Text":"Alphas 2x and Beta is nx."},{"Start":"00:56.465 ","End":"01:01.865","Text":"We get, and I\u0027m breaking it up into 2 integrals with the plus, we get this."},{"Start":"01:01.865 ","End":"01:03.800","Text":"We have 2x plus nx,"},{"Start":"01:03.800 ","End":"01:05.060","Text":"which is 2 plus nx,"},{"Start":"01:05.060 ","End":"01:07.490","Text":"and with the other one we have Alpha minus Beta,"},{"Start":"01:07.490 ","End":"01:12.215","Text":"which is 2x minus nx or 2 minus n times x."},{"Start":"01:12.215 ","End":"01:17.014","Text":"This is equal to the integral of sine is minus cosine,"},{"Start":"01:17.014 ","End":"01:23.771","Text":"and we have to divide by the 2 plus n and taken from 0 to Pi,"},{"Start":"01:23.771 ","End":"01:27.590","Text":"and I\u0027ve colored this minus with the 0 on the Pi,"},{"Start":"01:27.590 ","End":"01:31.295","Text":"meaning we can forget the minus if we switch the order here"},{"Start":"01:31.295 ","End":"01:35.495","Text":"of what we subtract from what the other one is similar."},{"Start":"01:35.495 ","End":"01:40.520","Text":"We get a plus here because again,"},{"Start":"01:40.520 ","End":"01:45.191","Text":"the integral of sine is minus cosine to the integral of minus sine is cosine."},{"Start":"01:45.191 ","End":"01:48.995","Text":"Now here we take first of all the 0,"},{"Start":"01:48.995 ","End":"01:51.680","Text":"which gives us 1 and subtract."},{"Start":"01:51.680 ","End":"01:53.825","Text":"What happens when you plug in Pi."},{"Start":"01:53.825 ","End":"02:01.940","Text":"Cosine of 2 plus nPi is minus 1^2 plus n. Here we get something similar."},{"Start":"02:01.940 ","End":"02:04.115","Text":"We just track this minus this."},{"Start":"02:04.115 ","End":"02:09.830","Text":"Well, I did 2 steps in 1 really should be the other way round the subtraction."},{"Start":"02:09.830 ","End":"02:12.860","Text":"But I reversed the order of the subtraction and also"},{"Start":"02:12.860 ","End":"02:16.315","Text":"reversed the 2 minus n. So it will be n minus 2,"},{"Start":"02:16.315 ","End":"02:18.785","Text":"and here, to match it,"},{"Start":"02:18.785 ","End":"02:23.255","Text":"I switched the order of 2 plus n. Okay, so this is what we get."},{"Start":"02:23.255 ","End":"02:25.490","Text":"I should\u0027ve included the middle step here."},{"Start":"02:25.490 ","End":"02:27.290","Text":"What we get here,"},{"Start":"02:27.290 ","End":"02:30.500","Text":"minus 1^2 plus n is minus 1^n,"},{"Start":"02:30.500 ","End":"02:33.320","Text":"minus 1^2, so it\u0027s just minus 1^n."},{"Start":"02:33.320 ","End":"02:35.735","Text":"I claim that these 2 are equal."},{"Start":"02:35.735 ","End":"02:38.750","Text":"If you divide 1 by the other,"},{"Start":"02:38.750 ","End":"02:42.400","Text":"then we get minus 1^2n, which is 1."},{"Start":"02:42.400 ","End":"02:44.705","Text":"So this is equal to this,"},{"Start":"02:44.705 ","End":"02:47.180","Text":"and this is equal to minus 1^n,"},{"Start":"02:47.180 ","End":"02:50.510","Text":"as we saw, the other 1 is also minus 1^n."},{"Start":"02:50.510 ","End":"02:52.370","Text":"If this equals this and this equals this,"},{"Start":"02:52.370 ","End":"02:53.990","Text":"and this equals this,"},{"Start":"02:53.990 ","End":"02:58.985","Text":"we can take this 1 minus minus 1^n outside the brackets."},{"Start":"02:58.985 ","End":"03:04.565","Text":"What we\u0027re left with is 1 over n plus 2 plus 1 over n minus 2."},{"Start":"03:04.565 ","End":"03:07.325","Text":"If we do this, addition,"},{"Start":"03:07.325 ","End":"03:12.740","Text":"comes out to be 2n over n^2 minus 4,"},{"Start":"03:12.740 ","End":"03:17.810","Text":"and I forgot to say earlier that this formula doesn\u0027t work when"},{"Start":"03:17.810 ","End":"03:23.796","Text":"n=2 because we don\u0027t want 0 in the denominator."},{"Start":"03:23.796 ","End":"03:26.765","Text":"Again here and is not equal to 2."},{"Start":"03:26.765 ","End":"03:30.230","Text":"We have to compute b_2 separately."},{"Start":"03:30.230 ","End":"03:32.540","Text":"b_2. If we go back,"},{"Start":"03:32.540 ","End":"03:41.285","Text":"it\u0027s the same thing as we had cosine 2x sine nx replacing n with 2 this time,"},{"Start":"03:41.285 ","End":"03:43.280","Text":"now we can use a trigonometric formula,"},{"Start":"03:43.280 ","End":"03:45.929","Text":"2 sine Alpha cosine Alpha sine 2 Alpha,"},{"Start":"03:45.929 ","End":"03:47.495","Text":"Alpha here is 2x."},{"Start":"03:47.495 ","End":"03:51.815","Text":"So this becomes sine of for x."},{"Start":"03:51.815 ","End":"03:54.004","Text":"Straightforward integral."},{"Start":"03:54.004 ","End":"03:56.243","Text":"Integral of sine is minus cosine,"},{"Start":"03:56.243 ","End":"04:00.350","Text":"and we have to divide by 4,"},{"Start":"04:00.350 ","End":"04:02.240","Text":"which I put together with the Pi."},{"Start":"04:02.240 ","End":"04:04.310","Text":"So we have the minus and the 4 here,"},{"Start":"04:04.310 ","End":"04:06.949","Text":"this between 0 and Pi."},{"Start":"04:06.949 ","End":"04:11.210","Text":"I claim this is 0 because if you put in Pi,"},{"Start":"04:11.210 ","End":"04:12.800","Text":"we get cosine of 4Pi,"},{"Start":"04:12.800 ","End":"04:15.380","Text":"put in 0 and cosine 4 times 0 is 0."},{"Start":"04:15.380 ","End":"04:20.555","Text":"Cosine of 4Pi is equal to cosine 0 because it\u0027s aperiodic to Pi."},{"Start":"04:20.555 ","End":"04:25.070","Text":"Well, you could also do it specifically and say that cosine of 4Pi"},{"Start":"04:25.070 ","End":"04:31.835","Text":"is 1 and cosine 0 is 1 event we get 0."},{"Start":"04:31.835 ","End":"04:37.680","Text":"So now we have b_n when it is not equal to 2 and we have b_2."},{"Start":"04:38.860 ","End":"04:43.385","Text":"Repeat that b_n is this, b_2 is this."},{"Start":"04:43.385 ","End":"04:48.275","Text":"Now we can write b_n separating evens and odds,"},{"Start":"04:48.275 ","End":"04:50.164","Text":"because when n is even,"},{"Start":"04:50.164 ","End":"04:52.475","Text":"this thing becomes 0."},{"Start":"04:52.475 ","End":"04:56.165","Text":"When n is odd, it\u0027s 1 minus minus 1, which is 2."},{"Start":"04:56.165 ","End":"05:00.770","Text":"So the 2 goes with the 2n and becomes 4n here,"},{"Start":"05:00.770 ","End":"05:02.765","Text":"this is even an odd,"},{"Start":"05:02.765 ","End":"05:09.185","Text":"and we can write it in terms of n=2k or n=2k minus 1."},{"Start":"05:09.185 ","End":"05:13.807","Text":"Here we plug in n=2k minus 1."},{"Start":"05:13.807 ","End":"05:20.149","Text":"Yeah, we don\u0027t need to separate the b_2 case anymore because b_2 is 0."},{"Start":"05:20.149 ","End":"05:23.120","Text":"Here when n=2k or n is even,"},{"Start":"05:23.120 ","End":"05:26.090","Text":"it\u0027s zeros, so it fits in with this pattern."},{"Start":"05:26.090 ","End":"05:30.395","Text":"So this is true for all n natural numbers."},{"Start":"05:30.395 ","End":"05:35.885","Text":"Now we have to plug into the formula f(x) is the sum b_n sine nx."},{"Start":"05:35.885 ","End":"05:37.850","Text":"So putting this n,"},{"Start":"05:37.850 ","End":"05:39.380","Text":"This is what we get."},{"Start":"05:39.380 ","End":"05:41.000","Text":"f(x) is cosine 2x,"},{"Start":"05:41.000 ","End":"05:42.575","Text":"so we have 1 over Pi."},{"Start":"05:42.575 ","End":"05:44.945","Text":"Well, say it out loud,"},{"Start":"05:44.945 ","End":"05:47.780","Text":"whatever is here, sine nx."},{"Start":"05:47.780 ","End":"05:51.900","Text":"That\u0027s the answer. We are done."}],"ID":28804}],"Thumbnail":null,"ID":294458},{"Name":"Summarizing Exercises","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1 - Part a","Duration":"3m 12s","ChapterTopicVideoID":27570,"CourseChapterTopicPlaylistID":294459,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"This exercise has 4 parts."},{"Start":"00:03.270 ","End":"00:06.855","Text":"In the first part we\u0027re given a function f(x)"},{"Start":"00:06.855 ","End":"00:12.060","Text":"equals e^i alpha x on the interval from minus Pi to Pi."},{"Start":"00:12.060 ","End":"00:15.150","Text":"Alpha is a real parameter,"},{"Start":"00:15.150 ","End":"00:18.240","Text":"but it\u0027s not a whole number, not an integer."},{"Start":"00:18.240 ","End":"00:21.870","Text":"If it was an integer the answer would be easy."},{"Start":"00:21.870 ","End":"00:22.965","Text":"If Alpha was say,"},{"Start":"00:22.965 ","End":"00:26.790","Text":"n then e^inx already is a Fourier series."},{"Start":"00:26.790 ","End":"00:30.900","Text":"One of the coefficients for c_n is 1 and all the rest of them are 0."},{"Start":"00:30.900 ","End":"00:33.420","Text":"So Alpha\u0027s not a whole number."},{"Start":"00:33.420 ","End":"00:35.280","Text":"Well, we\u0027ll start with Part A."},{"Start":"00:35.280 ","End":"00:37.880","Text":"We\u0027ll read each part as we come to it."},{"Start":"00:37.880 ","End":"00:44.903","Text":"In general, the formula for the complex Fourier series is a sum of c_ne^inx,"},{"Start":"00:44.903 ","End":"00:48.199","Text":"and the c_n is given by this formula."},{"Start":"00:48.199 ","End":"00:51.855","Text":"In our case, f(x) is e_i alpha x,"},{"Start":"00:51.855 ","End":"00:54.630","Text":"so let\u0027s compute c_n in our case."},{"Start":"00:54.630 ","End":"01:00.660","Text":"We have e^i alpha x is replacing f(x) by this,"},{"Start":"01:00.660 ","End":"01:02.700","Text":"e^minus inx dx,"},{"Start":"01:02.700 ","End":"01:07.650","Text":"and then using the rules of exponents i alpha x minus inx,"},{"Start":"01:07.650 ","End":"01:10.285","Text":"which is i Alpha minus nx."},{"Start":"01:10.285 ","End":"01:16.760","Text":"The integral of this is also an exponent that we have to divide by the coefficient of x,"},{"Start":"01:16.760 ","End":"01:18.905","Text":"which is i Alpha minus n,"},{"Start":"01:18.905 ","End":"01:24.270","Text":"so we\u0027re dividing by i alpha minus n. Note that Alpha is not a whole number,"},{"Start":"01:24.270 ","End":"01:27.990","Text":"so for any n, Alpha minus n is nonzero,"},{"Start":"01:27.990 ","End":"01:31.690","Text":"and we evaluate this from minus Pi to Pi."},{"Start":"01:31.690 ","End":"01:35.420","Text":"If x is Pi this is what we get."},{"Start":"01:35.420 ","End":"01:38.405","Text":"If x is minus Pi this is what we get."},{"Start":"01:38.405 ","End":"01:40.805","Text":"Now let\u0027s slightly rearrange it,"},{"Start":"01:40.805 ","End":"01:46.265","Text":"bring the Alpha minus n here and bring the 2 here."},{"Start":"01:46.265 ","End":"01:53.150","Text":"What we have is something of the form e^i Theta minus e^minus i Theta."},{"Start":"01:53.150 ","End":"01:54.800","Text":"We have the same thing here and here,"},{"Start":"01:54.800 ","End":"01:56.540","Text":"but with a minus."},{"Start":"01:56.540 ","End":"01:59.165","Text":"This is going to be sine Theta."},{"Start":"01:59.165 ","End":"02:02.160","Text":"This is 1 over Alpha minus n Pi,"},{"Start":"02:02.160 ","End":"02:03.690","Text":"that\u0027s in the denominator."},{"Start":"02:03.690 ","End":"02:06.540","Text":"Here we have sine of Theta,"},{"Start":"02:06.540 ","End":"02:09.555","Text":"which is Alpha minus n Pi."},{"Start":"02:09.555 ","End":"02:11.490","Text":"Now that we\u0027ve found c_n,"},{"Start":"02:11.490 ","End":"02:14.580","Text":"all we have to do is put it into the formula for"},{"Start":"02:14.580 ","End":"02:18.510","Text":"the Fourier series you just replace c_n by what we found it to be,"},{"Start":"02:18.510 ","End":"02:23.390","Text":"so f(x) is e^i alpha x is represented by"},{"Start":"02:23.390 ","End":"02:28.985","Text":"the sum of each of the inx here and the c_n just copied from here."},{"Start":"02:28.985 ","End":"02:31.070","Text":"But we can simplify this a bit."},{"Start":"02:31.070 ","End":"02:34.925","Text":"In general, this is the formula for sine Alpha minus Beta,"},{"Start":"02:34.925 ","End":"02:38.080","Text":"sine cosine minus cosine sine."},{"Start":"02:38.080 ","End":"02:42.875","Text":"What we have is Alpha Pi minus n Pi."},{"Start":"02:42.875 ","End":"02:45.575","Text":"In this formula, this is what we get."},{"Start":"02:45.575 ","End":"02:53.300","Text":"Note that cosine n Pi is minus 1^n and sine n Pi is 0,"},{"Start":"02:53.300 ","End":"02:57.580","Text":"so what we have is this times minus 1^n."},{"Start":"02:57.580 ","End":"02:59.975","Text":"Now let\u0027s apply this here."},{"Start":"02:59.975 ","End":"03:03.860","Text":"What we have is instead of sine of Alpha minus"},{"Start":"03:03.860 ","End":"03:08.945","Text":"n Pi we have sine of Alpha Pi times minus 1^n."},{"Start":"03:08.945 ","End":"03:12.900","Text":"This is the answer for Part A."}],"ID":28805},{"Watched":false,"Name":"Exercise 1 - Part b","Duration":"3m 3s","ChapterTopicVideoID":27571,"CourseChapterTopicPlaylistID":294459,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"Continuing with this exercise,"},{"Start":"00:02.700 ","End":"00:10.320","Text":"we come to part b where we have to show that this series equals this,"},{"Start":"00:10.320 ","End":"00:13.830","Text":"and we\u0027re going to use part a to show this."},{"Start":"00:13.830 ","End":"00:20.070","Text":"In part a, we showed that this is the Fourier series for e to the i Alpha x,"},{"Start":"00:20.070 ","End":"00:27.525","Text":"and we\u0027re going to apply Dirichlet\u0027s theorem which is pointwise convergence."},{"Start":"00:27.525 ","End":"00:32.400","Text":"Now e to the i Alpha x satisfies the conditions."},{"Start":"00:32.400 ","End":"00:37.124","Text":"It\u0027s piecewise continuous in fact it\u0027s continuous and"},{"Start":"00:37.124 ","End":"00:41.950","Text":"it has right and left derivatives at every point."},{"Start":"00:41.950 ","End":"00:44.655","Text":"We\u0027re going to be using the point x equals 0"},{"Start":"00:44.655 ","End":"00:49.385","Text":"and we\u0027re going to use the variation where f is continuous,"},{"Start":"00:49.385 ","End":"00:52.955","Text":"so meets the conditions at x equals naught."},{"Start":"00:52.955 ","End":"00:59.025","Text":"Plug in x equals naught and we get here 1 and here same,"},{"Start":"00:59.025 ","End":"01:03.600","Text":"I mean, e to the inx is 1 otherwise we have these coefficients."},{"Start":"01:03.600 ","End":"01:11.130","Text":"We have this, now we have to get it to look like what we had to show which is this."},{"Start":"01:11.130 ","End":"01:13.695","Text":"I\u0027ll do some algebra here."},{"Start":"01:13.695 ","End":"01:20.335","Text":"First of all, break the sum up into 3 sums from minus infinity to minus 1."},{"Start":"01:20.335 ","End":"01:23.854","Text":"Then the 0 term where n equals 0,"},{"Start":"01:23.854 ","End":"01:26.870","Text":"and then the sum from 1 to infinity."},{"Start":"01:26.870 ","End":"01:32.470","Text":"Here we can apply a change of index replacing n by minus n,"},{"Start":"01:32.470 ","End":"01:38.780","Text":"instead of going from minus infinity to minus 1 it goes from 1 to infinity,"},{"Start":"01:38.780 ","End":"01:42.395","Text":"and here we have minus 1 to the minus n,"},{"Start":"01:42.395 ","End":"01:46.595","Text":"which is actually the same as minus 1 to the n if you think about it."},{"Start":"01:46.595 ","End":"01:51.030","Text":"The exponents differ by 2n which is an even number which is"},{"Start":"01:51.030 ","End":"01:57.510","Text":"1 and n becomes plus n here and the term with the 0,"},{"Start":"01:57.510 ","End":"02:01.840","Text":"n equals 0 is just this and the last terms are the same."},{"Start":"02:01.840 ","End":"02:06.735","Text":"What we get, 1 is equal to this term."},{"Start":"02:06.735 ","End":"02:08.390","Text":"Here like I said,"},{"Start":"02:08.390 ","End":"02:10.540","Text":"we can replace the minus n with n,"},{"Start":"02:10.540 ","End":"02:12.410","Text":"it\u0027s the same thing."},{"Start":"02:12.410 ","End":"02:18.159","Text":"Now this and this have the same numerator and the same Pi in the denominator,"},{"Start":"02:18.159 ","End":"02:20.815","Text":"just Alpha plus n and Alpha minus n,"},{"Start":"02:20.815 ","End":"02:22.430","Text":"so we get this."},{"Start":"02:22.430 ","End":"02:29.505","Text":"Now we\u0027ll add this up 2 Alpha over Alpha squared minus n squared,"},{"Start":"02:29.505 ","End":"02:35.340","Text":"and here\u0027s the rest of it and I\u0027ve colored sine Alpha Pi over Pi."},{"Start":"02:35.340 ","End":"02:38.600","Text":"We\u0027re going to get rid of these by multiplying both sides by"},{"Start":"02:38.600 ","End":"02:42.235","Text":"the reciprocal which is Pi over sine Alpha Pi."},{"Start":"02:42.235 ","End":"02:45.240","Text":"Then we get Pi over sine Alpha Pi,"},{"Start":"02:45.240 ","End":"02:48.900","Text":"and then this part disappears as 1 over Alpha,"},{"Start":"02:48.900 ","End":"02:52.925","Text":"this part disappears so just this is what we have left."},{"Start":"02:52.925 ","End":"02:56.120","Text":"Next, we bring the 1 over Alpha to the other side and then switch"},{"Start":"02:56.120 ","End":"03:01.220","Text":"sides and we get this and this is what we had to show."},{"Start":"03:01.220 ","End":"03:04.170","Text":"That concludes part b."}],"ID":28806},{"Watched":false,"Name":"Exercise 1 - Part c","Duration":"1m 40s","ChapterTopicVideoID":27572,"CourseChapterTopicPlaylistID":294459,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:03.585","Text":"We just finished parts a and b,"},{"Start":"00:03.585 ","End":"00:06.105","Text":"and now we\u0027re heading for part c,"},{"Start":"00:06.105 ","End":"00:12.255","Text":"where we have to show this series sums to this."},{"Start":"00:12.255 ","End":"00:15.720","Text":"In part b, we use the Fourier\u0027s slaves theorem in part c,"},{"Start":"00:15.720 ","End":"00:18.645","Text":"we\u0027re going to be using Parseval\u0027s identity."},{"Start":"00:18.645 ","End":"00:23.165","Text":"This is the Fourier series we found in part a."},{"Start":"00:23.165 ","End":"00:26.060","Text":"We\u0027re going to apply Parseval\u0027s identity."},{"Start":"00:26.060 ","End":"00:28.310","Text":"I didn\u0027t quote it again,"},{"Start":"00:28.310 ","End":"00:29.800","Text":"go back and look it up."},{"Start":"00:29.800 ","End":"00:34.729","Text":"1 over 2 Pi the integral of f(x) absolute value squared dx,"},{"Start":"00:34.729 ","End":"00:36.035","Text":"which is just this,"},{"Start":"00:36.035 ","End":"00:41.285","Text":"is equal to the sum of absolute value of the coefficient squared."},{"Start":"00:41.285 ","End":"00:43.685","Text":"Now, the absolute value of e to the i,"},{"Start":"00:43.685 ","End":"00:46.460","Text":"something real to point on the unit circle,"},{"Start":"00:46.460 ","End":"00:48.110","Text":"it\u0027s absolute value is 1,"},{"Start":"00:48.110 ","End":"00:50.395","Text":"and squared is still 1."},{"Start":"00:50.395 ","End":"00:53.930","Text":"Here, this is a real quantity,"},{"Start":"00:53.930 ","End":"00:56.690","Text":"so we don\u0027t need the absolute value when you square it,"},{"Start":"00:56.690 ","End":"01:03.290","Text":"-1 to the n drops off and we\u0027re left with this."},{"Start":"01:03.290 ","End":"01:05.210","Text":"Everything\u0027s squared. This is squared,"},{"Start":"01:05.210 ","End":"01:07.400","Text":"this is squared, this is squared."},{"Start":"01:07.400 ","End":"01:10.070","Text":"Now, we can do the integral."},{"Start":"01:10.070 ","End":"01:12.410","Text":"The integral of this is 2Pi,"},{"Start":"01:12.410 ","End":"01:14.905","Text":"but divided by 2Pi is just 1."},{"Start":"01:14.905 ","End":"01:18.820","Text":"Here, we can pull sine squared Alpha Pi over Pi"},{"Start":"01:18.820 ","End":"01:22.648","Text":"squared in front of the sum because they don\u0027t contain n,"},{"Start":"01:22.648 ","End":"01:24.125","Text":"and so we have this."},{"Start":"01:24.125 ","End":"01:26.390","Text":"Now, just bringing this to"},{"Start":"01:26.390 ","End":"01:30.803","Text":"the other side we\u0027re multiplying both sides by the reciprocal,"},{"Start":"01:30.803 ","End":"01:31.880","Text":"and then switching sides,"},{"Start":"01:31.880 ","End":"01:35.930","Text":"we get that this upside-down is this equals this."},{"Start":"01:35.930 ","End":"01:37.820","Text":"That\u0027s what we had to show,"},{"Start":"01:37.820 ","End":"01:40.560","Text":"so that concludes part c."}],"ID":28807},{"Watched":false,"Name":"Exercise 1 - Part d","Duration":"8m 59s","ChapterTopicVideoID":27573,"CourseChapterTopicPlaylistID":294459,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:03.330","Text":"Now we come to the last of the 4 parts."},{"Start":"00:03.330 ","End":"00:07.020","Text":"We come to part d where we have to show that,"},{"Start":"00:07.020 ","End":"00:10.965","Text":"well, this is complicated to read if you look at it."},{"Start":"00:10.965 ","End":"00:15.690","Text":"Just to remind you that secant is 1 over cosine."},{"Start":"00:15.690 ","End":"00:19.740","Text":"We could have said 1 over the cosine of Alpha Pi over 2."},{"Start":"00:19.740 ","End":"00:21.810","Text":"This is the result of Part A,"},{"Start":"00:21.810 ","End":"00:25.095","Text":"the Fourier series for e^i Alpha x."},{"Start":"00:25.095 ","End":"00:28.140","Text":"We\u0027re going to break this up into 3 sums,"},{"Start":"00:28.140 ","End":"00:30.330","Text":"we did something similar earlier."},{"Start":"00:30.330 ","End":"00:32.850","Text":"From minus infinity to minus 1,"},{"Start":"00:32.850 ","End":"00:36.255","Text":"just the 0 term and then from 1 to infinity."},{"Start":"00:36.255 ","End":"00:39.090","Text":"This one we make a change of index,"},{"Start":"00:39.090 ","End":"00:43.680","Text":"replace n by minus n. Here we get Alpha plus n,"},{"Start":"00:43.680 ","End":"00:46.860","Text":"and here we get e^minus inx."},{"Start":"00:46.860 ","End":"00:48.690","Text":"Here we get a minus,"},{"Start":"00:48.690 ","End":"00:54.800","Text":"but we could throw this minus out because minus 1^minus n is the same as minus 1^n,"},{"Start":"00:54.800 ","End":"00:56.880","Text":"we\u0027ve seen this before."},{"Start":"00:57.700 ","End":"01:02.090","Text":"Anyway, we can collect what similar in the first and last term."},{"Start":"01:02.090 ","End":"01:04.670","Text":"In other words, in these sums."},{"Start":"01:04.670 ","End":"01:12.620","Text":"Let\u0027s see, this is the same as this because we said minus 3,"},{"Start":"01:12.620 ","End":"01:13.865","Text":"it can be thrown out."},{"Start":"01:13.865 ","End":"01:17.689","Text":"Also, we have a Pi here and a Pi here."},{"Start":"01:17.689 ","End":"01:21.065","Text":"When we combine this sum with this sum,"},{"Start":"01:21.065 ","End":"01:25.075","Text":"this part can go in front."},{"Start":"01:25.075 ","End":"01:33.780","Text":"All we\u0027re left with then is e^inx over Alpha plus n and e^minus inx,"},{"Start":"01:33.780 ","End":"01:39.870","Text":"and then e^inx over Alpha minus n. What we\u0027re going to do next is"},{"Start":"01:39.870 ","End":"01:46.235","Text":"work on this sum and try to simplify it or modify it to make it more useful."},{"Start":"01:46.235 ","End":"01:49.115","Text":"In the next few lines, we\u0027re just going to be working on this part."},{"Start":"01:49.115 ","End":"01:54.270","Text":"First thing we can do is just cross-multiply and the denominator,"},{"Start":"01:54.270 ","End":"01:56.400","Text":"Alpha squared minus n squared."},{"Start":"01:56.400 ","End":"01:58.710","Text":"I collect the terms with Alpha,"},{"Start":"01:58.710 ","End":"02:03.615","Text":"it\u0027s Alpha times e^inx plus e^minus inx."},{"Start":"02:03.615 ","End":"02:05.460","Text":"Then the parts with n,"},{"Start":"02:05.460 ","End":"02:09.180","Text":"so it\u0027s e^inx and then,"},{"Start":"02:09.180 ","End":"02:11.715","Text":"all the plus should be a minus, sorry,"},{"Start":"02:11.715 ","End":"02:15.795","Text":"that\u0027s the e^minus inx with the minus n. Here I"},{"Start":"02:15.795 ","End":"02:19.870","Text":"also took some constant in front of the sum,"},{"Start":"02:19.870 ","End":"02:22.180","Text":"the sine Alpha Pi over Pi."},{"Start":"02:22.180 ","End":"02:26.680","Text":"Next thing we can do is use the formulas."},{"Start":"02:26.680 ","End":"02:29.410","Text":"There are 2 formulas I just wrote 1 of them,"},{"Start":"02:29.410 ","End":"02:34.735","Text":"the sine Theta and for cosine Theta sine has a minus here and an i here."},{"Start":"02:34.735 ","End":"02:36.055","Text":"If we make it cosine,"},{"Start":"02:36.055 ","End":"02:39.020","Text":"it\u0027s a plus here and there is no i here."},{"Start":"02:39.020 ","End":"02:41.745","Text":"Theta is nx here,"},{"Start":"02:41.745 ","End":"02:45.700","Text":"so we get sine nx and there\u0027s a 2 i here."},{"Start":"02:45.700 ","End":"02:48.440","Text":"For the cosine, like I said, it\u0027s a 2,"},{"Start":"02:48.440 ","End":"02:53.130","Text":"this plus this is cosine of Theta, which is nx."},{"Start":"02:53.130 ","End":"02:56.900","Text":"Next, what we\u0027re going to do is take the real part of both sides."},{"Start":"02:56.900 ","End":"03:00.620","Text":"Now if I break this up into real and imaginary,"},{"Start":"03:00.620 ","End":"03:03.640","Text":"it\u0027s cosine Alpha x is the real part."},{"Start":"03:03.640 ","End":"03:07.370","Text":"On the right side, this is real,"},{"Start":"03:07.370 ","End":"03:09.020","Text":"this constant is real,"},{"Start":"03:09.020 ","End":"03:11.275","Text":"so we just have to break this up."},{"Start":"03:11.275 ","End":"03:13.175","Text":"There\u0027s an i here,"},{"Start":"03:13.175 ","End":"03:18.440","Text":"and this is where we break it up into the real part and the imaginary part."},{"Start":"03:18.440 ","End":"03:22.070","Text":"We\u0027re going to compare the real with the real,"},{"Start":"03:22.070 ","End":"03:23.660","Text":"this is also real."},{"Start":"03:23.660 ","End":"03:27.975","Text":"What we get is cosine Alpha x has the sum,"},{"Start":"03:27.975 ","End":"03:33.825","Text":"this plus this times just this part."},{"Start":"03:33.825 ","End":"03:37.200","Text":"I brought the cosine nx over here to the end."},{"Start":"03:37.200 ","End":"03:39.860","Text":"Now going back to the exercise,"},{"Start":"03:39.860 ","End":"03:42.395","Text":"if we look at the denominator here,"},{"Start":"03:42.395 ","End":"03:47.405","Text":"if you think of 2n plus 1 is just n probably would just taking odd numbers."},{"Start":"03:47.405 ","End":"03:52.220","Text":"This looks like n times n^2 minus Alpha^2."},{"Start":"03:52.220 ","End":"03:57.450","Text":"But back here, we have n^2 minus Alpha^2,"},{"Start":"03:57.450 ","End":"04:00.610","Text":"there\u0027s no extra n. Using the Fourier series,"},{"Start":"04:00.610 ","End":"04:02.660","Text":"no matter what you substitute for x,"},{"Start":"04:02.660 ","End":"04:06.350","Text":"you won\u0027t get the extra n. The way we can get"},{"Start":"04:06.350 ","End":"04:10.175","Text":"that is by doing an integration term by term,"},{"Start":"04:10.175 ","End":"04:12.800","Text":"we can either do it as an indefinite integral,"},{"Start":"04:12.800 ","End":"04:15.530","Text":"then we have to adjust the constant,"},{"Start":"04:15.530 ","End":"04:23.915","Text":"or we can integrate each term from 0 to t and get a function of t. Let\u0027s see what we get."},{"Start":"04:23.915 ","End":"04:28.220","Text":"Cosine Alpha x becomes sine Alpha x,"},{"Start":"04:28.220 ","End":"04:30.050","Text":"but between 0 and t,"},{"Start":"04:30.050 ","End":"04:33.955","Text":"It comes out sine Alpha t over and Alpha here."},{"Start":"04:33.955 ","End":"04:38.900","Text":"Here again, the cosine nx becomes"},{"Start":"04:38.900 ","End":"04:46.010","Text":"sine nx between 0 and t is sine nt minus sine 0."},{"Start":"04:46.010 ","End":"04:49.760","Text":"Because the integral of cosine nx is sine of x over n,"},{"Start":"04:49.760 ","End":"04:51.455","Text":"and we get this n here."},{"Start":"04:51.455 ","End":"04:53.960","Text":"We might get what we want."},{"Start":"04:53.960 ","End":"04:57.490","Text":"Now we want to substitute something."},{"Start":"04:57.490 ","End":"05:02.795","Text":"When we take the integral of a Fourier series and we always can,"},{"Start":"05:02.795 ","End":"05:07.250","Text":"then we get a series that is uniformly convergent."},{"Start":"05:07.250 ","End":"05:12.800","Text":"It may not be a Fourier series because we can get a linear term like this,"},{"Start":"05:12.800 ","End":"05:15.980","Text":"but it\u0027s always uniformly convergent"},{"Start":"05:15.980 ","End":"05:19.475","Text":"and uniform convergence implies pointwise convergence."},{"Start":"05:19.475 ","End":"05:25.645","Text":"We can substitute t equals Pi over 2, and that\u0027s what we\u0027ll do."},{"Start":"05:25.645 ","End":"05:31.870","Text":"This t becomes Pi over 2 and this t becomes Pi over 2."},{"Start":"05:31.870 ","End":"05:36.395","Text":"But the thing is that when n is even, this is 0."},{"Start":"05:36.395 ","End":"05:39.445","Text":"We really only need the odd n\u0027s."},{"Start":"05:39.445 ","End":"05:41.445","Text":"If n is 2k,"},{"Start":"05:41.445 ","End":"05:46.410","Text":"then 2k times Pi over 2 is k Pi and sine k Pi is 0."},{"Start":"05:46.410 ","End":"05:50.745","Text":"We\u0027re just going to sum over the 2k plus 1,"},{"Start":"05:50.745 ","End":"05:54.225","Text":"so we get this equals."},{"Start":"05:54.225 ","End":"05:59.835","Text":"Instead of sine n Pi over 2 replace n by 2k plus 1."},{"Start":"05:59.835 ","End":"06:03.495","Text":"Then we get k Pi plus Pi over 2."},{"Start":"06:03.495 ","End":"06:06.330","Text":"Here instead of n 2k plus 1."},{"Start":"06:06.330 ","End":"06:10.570","Text":"Now we can use trigonometric identities on this."},{"Start":"06:10.570 ","End":"06:14.160","Text":"If we just look at the sine of k Pi plus Pi over 2."},{"Start":"06:14.160 ","End":"06:15.830","Text":"There\u0027s many number of ways we can do it,"},{"Start":"06:15.830 ","End":"06:17.870","Text":"I\u0027m using the formula for sine of a sum,"},{"Start":"06:17.870 ","End":"06:22.615","Text":"sine Alpha plus Beta is sine Alpha cosine Beta plus cosine Alpha sine Beta."},{"Start":"06:22.615 ","End":"06:25.470","Text":"This part is 0, this part is 1."},{"Start":"06:25.470 ","End":"06:28.010","Text":"We just have the cosine k Pi,"},{"Start":"06:28.010 ","End":"06:30.455","Text":"which is minus 1^k."},{"Start":"06:30.455 ","End":"06:34.190","Text":"We can replace this by minus 1^k,"},{"Start":"06:34.190 ","End":"06:37.235","Text":"and everything else is the same."},{"Start":"06:37.235 ","End":"06:39.725","Text":"Now to get from here to here,"},{"Start":"06:39.725 ","End":"06:46.025","Text":"what I did was multiply both sides by the upside down of this,"},{"Start":"06:46.025 ","End":"06:48.575","Text":"Pi over sine Alpha Pi."},{"Start":"06:48.575 ","End":"06:50.240","Text":"If we do that,"},{"Start":"06:50.240 ","End":"06:58.970","Text":"this and this will disappear from this term and we\u0027ll just have Pi over 2 Alpha."},{"Start":"06:58.970 ","End":"07:08.415","Text":"Here we get an extra Pi and sine Alpha Pi and the rest of its like here."},{"Start":"07:08.415 ","End":"07:11.130","Text":"This disappears over here."},{"Start":"07:11.130 ","End":"07:15.815","Text":"The next step will be to divide both sides by 2 Alpha,"},{"Start":"07:15.815 ","End":"07:18.415","Text":"the 2 Alpha will disappear from here."},{"Start":"07:18.415 ","End":"07:20.400","Text":"We get an extra 2 Alpha here,"},{"Start":"07:20.400 ","End":"07:22.470","Text":"making it 4 Alpha^2,"},{"Start":"07:22.470 ","End":"07:24.000","Text":"an extra 2 Alpha here,"},{"Start":"07:24.000 ","End":"07:26.435","Text":"so it\u0027s 2 Alpha^2 here."},{"Start":"07:26.435 ","End":"07:30.505","Text":"Now, just flip side switch left and right."},{"Start":"07:30.505 ","End":"07:33.260","Text":"Also to get rid of the minus,"},{"Start":"07:33.260 ","End":"07:36.815","Text":"just have to switch the order of these 2."},{"Start":"07:36.815 ","End":"07:39.860","Text":"This is the reverse order of this,"},{"Start":"07:39.860 ","End":"07:41.660","Text":"the minus is gone."},{"Start":"07:41.660 ","End":"07:47.270","Text":"Now we\u0027re going to do some simplification for just this part here."},{"Start":"07:47.270 ","End":"07:51.815","Text":"Pi sine Alpha Pi over 2 over 2 Alpha squared sine Alpha Pi."},{"Start":"07:51.815 ","End":"07:57.050","Text":"We can use a trigonometric identity for the double angle and say that"},{"Start":"07:57.050 ","End":"08:03.145","Text":"sine Alpha Pi is 2 sine Alpha Pi over 2 cosine Alpha Pi over 2."},{"Start":"08:03.145 ","End":"08:07.080","Text":"Sine of 2 Theta is 2 sine Theta cosine Theta,"},{"Start":"08:07.080 ","End":"08:09.510","Text":"and Theta would behalf of this."},{"Start":"08:09.510 ","End":"08:15.450","Text":"Then the sine Alpha Pi over 2 and the sine Alpha Pi over 2 cancels."},{"Start":"08:15.450 ","End":"08:20.350","Text":"We can bring the cosine Alpha Pi over 2 to the numerator if we call it secant,"},{"Start":"08:20.350 ","End":"08:22.550","Text":"1 over cosine is secant."},{"Start":"08:22.550 ","End":"08:26.330","Text":"What we\u0027re left with here is 2 Alpha squared times 2 is 4 Alpha squared,"},{"Start":"08:26.330 ","End":"08:29.605","Text":"and here Pi secant Alpha Pi over 2."},{"Start":"08:29.605 ","End":"08:33.830","Text":"Next we can replace this with this."},{"Start":"08:33.830 ","End":"08:40.745","Text":"That the sum here just the same except instead of this, we\u0027re writing this."},{"Start":"08:40.745 ","End":"08:45.710","Text":"Next note that we have Pi over 4 Alpha squared and Pi over 4 Alpha squared."},{"Start":"08:45.710 ","End":"08:48.215","Text":"We can take that outside the brackets."},{"Start":"08:48.215 ","End":"08:52.445","Text":"What we\u0027re left with a secant Alpha Pi over 2 minus 1."},{"Start":"08:52.445 ","End":"08:55.805","Text":"This is the expression we had to arrive at."},{"Start":"08:55.805 ","End":"09:00.390","Text":"That concludes part d and all of this exercise."}],"ID":28808},{"Watched":false,"Name":"Exercise 2 - Part a","Duration":"4m 4s","ChapterTopicVideoID":27574,"CourseChapterTopicPlaylistID":294459,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.270","Text":"In this exercise, f(x) equals absolute value of x on the interval from minus Pi to Pi,"},{"Start":"00:09.270 ","End":"00:14.159","Text":"and f\u0027(x) as its derivative,"},{"Start":"00:14.159 ","End":"00:16.551","Text":"well, wherever the derivative exists."},{"Start":"00:16.551 ","End":"00:20.505","Text":"The derivative doesn\u0027t exist at x=0, for example,"},{"Start":"00:20.505 ","End":"00:22.250","Text":"but generally it does,"},{"Start":"00:22.250 ","End":"00:25.075","Text":"at least, it has a piece-wise derivative."},{"Start":"00:25.075 ","End":"00:28.665","Text":"In Part A, we have to find the real Fourier series of f,"},{"Start":"00:28.665 ","End":"00:30.810","Text":"and we\u0027ll read the other parts as we come to them."},{"Start":"00:30.810 ","End":"00:32.775","Text":"We\u0027re starting with Part A."},{"Start":"00:32.775 ","End":"00:39.615","Text":"This is a general formula for the Fourier series of f is of this form where,"},{"Start":"00:39.615 ","End":"00:43.950","Text":"a_n and b_n are given by these formulas."},{"Start":"00:43.950 ","End":"00:47.670","Text":"In our case, f(x) is absolute value of x,"},{"Start":"00:47.670 ","End":"00:49.370","Text":"and that\u0027s an even function."},{"Start":"00:49.370 ","End":"00:51.139","Text":"When we have an even function,"},{"Start":"00:51.139 ","End":"00:52.520","Text":"the b_n is a 0,"},{"Start":"00:52.520 ","End":"00:54.860","Text":"and we just have a sum of cosines,"},{"Start":"00:54.860 ","End":"00:56.825","Text":"and a_n is, well,"},{"Start":"00:56.825 ","End":"01:01.150","Text":"this formula, but replace f(x) by absolute value of x."},{"Start":"01:01.150 ","End":"01:03.200","Text":"Now, let\u0027s do the computation."},{"Start":"01:03.200 ","End":"01:04.970","Text":"Let\u0027s compute a_n."},{"Start":"01:04.970 ","End":"01:12.905","Text":"Because it\u0027s even, we can double it and just take half the interval from 0-Pi."},{"Start":"01:12.905 ","End":"01:16.340","Text":"We can also then replace absolute value of x by x,"},{"Start":"01:16.340 ","End":"01:19.640","Text":"because that\u0027s what it is on the positive side."},{"Start":"01:19.640 ","End":"01:23.345","Text":"We\u0027ll do this with integration by parts."},{"Start":"01:23.345 ","End":"01:25.445","Text":"This is the formula."},{"Start":"01:25.445 ","End":"01:28.885","Text":"We\u0027ll take this as u, this is v\u0027."},{"Start":"01:28.885 ","End":"01:33.980","Text":"We need uv minus the integral of u\u0027v."},{"Start":"01:33.980 ","End":"01:36.607","Text":"This is the integral we get."},{"Start":"01:36.607 ","End":"01:39.800","Text":"But notice that this doesn\u0027t work when n is 0."},{"Start":"01:39.800 ","End":"01:41.840","Text":"We\u0027ll have to do that separately."},{"Start":"01:41.840 ","End":"01:45.270","Text":"That\u0027s often the case that n equals 0 is a special case."},{"Start":"01:45.270 ","End":"01:46.580","Text":"This is equal 2 now."},{"Start":"01:46.580 ","End":"01:55.245","Text":"This part is 0 because sine of n times 0 or sine of n times Pi are both 0,"},{"Start":"01:55.245 ","End":"02:00.240","Text":"and we\u0027re just left with this integral."},{"Start":"02:00.240 ","End":"02:05.300","Text":"Just take the 2/Pi in front and the n in front and the minus"},{"Start":"02:05.300 ","End":"02:10.385","Text":"in front so we get minus 2Pi integral of sine (nx)dx."},{"Start":"02:10.385 ","End":"02:12.590","Text":"This is a straightforward integral."},{"Start":"02:12.590 ","End":"02:16.115","Text":"The integral of minus sine is cosine,"},{"Start":"02:16.115 ","End":"02:20.795","Text":"and we have to divide by another n. That makes this n^2."},{"Start":"02:20.795 ","End":"02:24.380","Text":"Now, if you plug in n equals 0 and Pi,"},{"Start":"02:24.380 ","End":"02:27.480","Text":"then we get cosine nPi,"},{"Start":"02:27.480 ","End":"02:29.070","Text":"which is minus 1^n,"},{"Start":"02:29.070 ","End":"02:30.885","Text":"cosine of 0 is 1."},{"Start":"02:30.885 ","End":"02:32.475","Text":"This is the expression we get."},{"Start":"02:32.475 ","End":"02:35.000","Text":"Obviously, it depends on whether n is odd or even."},{"Start":"02:35.000 ","End":"02:37.810","Text":"This is either 1 or minus 1."},{"Start":"02:37.810 ","End":"02:41.490","Text":"If n is 2k, or even,"},{"Start":"02:41.490 ","End":"02:43.830","Text":"this is 1 minus 1 is 0."},{"Start":"02:43.830 ","End":"02:45.495","Text":"If n is odd,"},{"Start":"02:45.495 ","End":"02:47.430","Text":"or 2k minus 1,"},{"Start":"02:47.430 ","End":"02:53.190","Text":"then this is minus 1 minus 1 is minus 2 times 2 is minus 4."},{"Start":"02:53.190 ","End":"02:58.645","Text":"Pi(n)^2 squared is Pi(2k minus 1)^2 because that is 2k minus 1."},{"Start":"02:58.645 ","End":"03:01.835","Text":"That\u0027s for n not equal to 0."},{"Start":"03:01.835 ","End":"03:09.120","Text":"If n equals 0, we compute directly a_0 is 2/Pi integral xdx,"},{"Start":"03:09.120 ","End":"03:11.790","Text":"integral of x is a 1/2x^2."},{"Start":"03:11.790 ","End":"03:16.395","Text":"Pi^2/2 times 2/Pi comes out to be just Pi."},{"Start":"03:16.395 ","End":"03:17.685","Text":"Let\u0027s summarize."},{"Start":"03:17.685 ","End":"03:20.850","Text":"Yeah, a_0 is Pi, and a_0,"},{"Start":"03:20.850 ","End":"03:23.730","Text":"if n is not 0, is this,"},{"Start":"03:23.730 ","End":"03:25.475","Text":"okay, we already said this."},{"Start":"03:25.475 ","End":"03:29.314","Text":"Now, recall the general formula for"},{"Start":"03:29.314 ","End":"03:34.780","Text":"an even functions for Reyes series as a cosine series."},{"Start":"03:34.780 ","End":"03:39.375","Text":"Just plug in, we have a_0 here and we have a_n here."},{"Start":"03:39.375 ","End":"03:43.590","Text":"We get a_0/2 is Pi/2,"},{"Start":"03:43.590 ","End":"03:47.325","Text":"and a_n, well,"},{"Start":"03:47.325 ","End":"03:50.205","Text":"replace n by 2k minus 1,"},{"Start":"03:50.205 ","End":"03:53.160","Text":"and then also k goes from 1 to infinity,"},{"Start":"03:53.160 ","End":"03:58.980","Text":"so it\u0027s this here times cosine nx,"},{"Start":"03:58.980 ","End":"04:01.095","Text":"which is ([2k minus 1]x)."},{"Start":"04:01.095 ","End":"04:04.240","Text":"This concludes Part A."}],"ID":28809},{"Watched":false,"Name":"Exercise 2 - Part b","Duration":"5m 13s","ChapterTopicVideoID":27565,"CourseChapterTopicPlaylistID":294459,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.635","Text":"We just did Part A and now we come to Part B."},{"Start":"00:04.635 ","End":"00:11.985","Text":"We have to see what the following series converges to on the whole real line."},{"Start":"00:11.985 ","End":"00:14.520","Text":"This is the answer for Part A,"},{"Start":"00:14.520 ","End":"00:20.325","Text":"the cosine series of absolute value of x is the following."},{"Start":"00:20.325 ","End":"00:24.645","Text":"Here\u0027s a picture of function absolute value of x."},{"Start":"00:24.645 ","End":"00:28.770","Text":"What you see in faint here is the periodic extension."},{"Start":"00:28.770 ","End":"00:30.960","Text":"We\u0027ll come to that, not yet."},{"Start":"00:30.960 ","End":"00:37.730","Text":"Now, we want to do a term-by-term differentiation to get f prime of x,"},{"Start":"00:37.730 ","End":"00:40.820","Text":"and we can do that because it meets the conditions."},{"Start":"00:40.820 ","End":"00:43.565","Text":"To summarize the conditions, there are 3 of them."},{"Start":"00:43.565 ","End":"00:46.280","Text":"F has to be continuous on the interval,"},{"Start":"00:46.280 ","End":"00:48.035","Text":"and it certainly is,"},{"Start":"00:48.035 ","End":"00:50.960","Text":"f of minus Pi equals f(Pi)."},{"Start":"00:50.960 ","End":"00:54.870","Text":"Yes, they\u0027re both equal to Pi here and here."},{"Start":"00:54.920 ","End":"00:59.975","Text":"The derivative has to be piecewise continuous."},{"Start":"00:59.975 ","End":"01:04.040","Text":"The derivative is minus 1 here,"},{"Start":"01:04.040 ","End":"01:05.480","Text":"and 1 here,"},{"Start":"01:05.480 ","End":"01:08.150","Text":"and it\u0027s not defined at this point,"},{"Start":"01:08.150 ","End":"01:10.610","Text":"but at least on the open interval minus Pi, 0,"},{"Start":"01:10.610 ","End":"01:13.490","Text":"on the open interval 0, Pi it is continuous."},{"Start":"01:13.490 ","End":"01:17.399","Text":"Show you a picture of the derivative."},{"Start":"01:17.399 ","End":"01:20.660","Text":"Like we said that here it\u0027s 1 and here it\u0027s minus 1."},{"Start":"01:20.660 ","End":"01:23.600","Text":"It\u0027s not defined at 0 or at the endpoints,"},{"Start":"01:23.600 ","End":"01:25.565","Text":"but it\u0027s piecewise continuous."},{"Start":"01:25.565 ","End":"01:28.550","Text":"We can write a formula for this function which start with"},{"Start":"01:28.550 ","End":"01:32.735","Text":"f(x) as the absolute value of x, which is this."},{"Start":"01:32.735 ","End":"01:34.910","Text":"Then if you differentiate,"},{"Start":"01:34.910 ","End":"01:38.180","Text":"then you get either 1 or minus 1,"},{"Start":"01:38.180 ","End":"01:42.385","Text":"and this has a name, it\u0027s the sign function sign,"},{"Start":"01:42.385 ","End":"01:45.330","Text":"S-I-G-N, 1 or minus 1."},{"Start":"01:45.330 ","End":"01:48.620","Text":"Now let\u0027s do the term by term differentiation."},{"Start":"01:48.620 ","End":"01:52.865","Text":"This is the series for f(x) that we had,"},{"Start":"01:52.865 ","End":"01:58.200","Text":"so f prime of x will be gotten by term by term differentiation."},{"Start":"01:58.200 ","End":"01:59.970","Text":"The Pi over 2 disappears,"},{"Start":"01:59.970 ","End":"02:04.120","Text":"derivative of cosine is minus sine,"},{"Start":"02:04.120 ","End":"02:07.400","Text":"the minus and the minus cancel."},{"Start":"02:07.400 ","End":"02:10.705","Text":"But we also have the inner derivative 2k minus 1,"},{"Start":"02:10.705 ","End":"02:16.050","Text":"which makes this 2k minus 1 squared become just 2k minus 1."},{"Start":"02:16.050 ","End":"02:20.900","Text":"Now let\u0027s bring the 4 and the Pi to the other side."},{"Start":"02:20.900 ","End":"02:23.705","Text":"Then we get Pi over 4 here,"},{"Start":"02:23.705 ","End":"02:25.285","Text":"and here we get the sum,"},{"Start":"02:25.285 ","End":"02:27.405","Text":"I can put this on the numerator,"},{"Start":"02:27.405 ","End":"02:30.645","Text":"sine 2k minus 1 over 2k minus 1."},{"Start":"02:30.645 ","End":"02:33.870","Text":"If you let k equals 1,2,3,4,5,"},{"Start":"02:33.870 ","End":"02:35.835","Text":"we get sine x,"},{"Start":"02:35.835 ","End":"02:38.025","Text":"1/3 sine 3x,1/5 sine 5x,"},{"Start":"02:38.025 ","End":"02:40.555","Text":"1/7 sine 7x and so on."},{"Start":"02:40.555 ","End":"02:45.575","Text":"We have the sum of this is Pi over 4 f prime of x."},{"Start":"02:45.575 ","End":"02:49.985","Text":"Now f prime of x we showed was either 1 or minus 1."},{"Start":"02:49.985 ","End":"02:52.670","Text":"Then we can just multiply it by Pi over 4,"},{"Start":"02:52.670 ","End":"02:55.720","Text":"so it\u0027s Pi over 4 or minus Pi over 4."},{"Start":"02:55.720 ","End":"03:00.885","Text":"This series is equal to Pi over 4 f prime of x,"},{"Start":"03:00.885 ","End":"03:04.285","Text":"which is Pi over 4 or minus Pi over 4,"},{"Start":"03:04.285 ","End":"03:05.915","Text":"or something else in a moment."},{"Start":"03:05.915 ","End":"03:10.015","Text":"Let\u0027s call this function g(x)."},{"Start":"03:10.015 ","End":"03:13.230","Text":"I claim that it\u0027s 0,"},{"Start":"03:13.230 ","End":"03:15.975","Text":"when x is 0 or at the endpoints."},{"Start":"03:15.975 ","End":"03:18.424","Text":"We\u0027ll use the Dirichlet theorem."},{"Start":"03:18.424 ","End":"03:21.350","Text":"At x equals 0, the series converges"},{"Start":"03:21.350 ","End":"03:26.315","Text":"to the average and the limit from the right and the limit from the left,"},{"Start":"03:26.315 ","End":"03:32.720","Text":"so it\u0027s Pi over 4 minus Pi over 4 divided by 2, which is 0."},{"Start":"03:32.720 ","End":"03:34.370","Text":"Also at plus and minus"},{"Start":"03:34.370 ","End":"03:40.790","Text":"Pi the Dirichlet theorem says that it converges to the average at the 2 endpoints,"},{"Start":"03:40.790 ","End":"03:44.015","Text":"minus Pi on the right and plus Pi on the left,"},{"Start":"03:44.015 ","End":"03:46.190","Text":"and again we get 0."},{"Start":"03:46.190 ","End":"03:49.435","Text":"That explains this part here."},{"Start":"03:49.435 ","End":"03:52.880","Text":"Now, the Fourier series converges to"},{"Start":"03:52.880 ","End":"03:59.705","Text":"the 2 Pi periodic extension of g. Call that g wave g Tilde."},{"Start":"03:59.705 ","End":"04:02.380","Text":"Let\u0027s go back to the picture we had."},{"Start":"04:02.380 ","End":"04:06.720","Text":"Here it is, and we can do the extension."},{"Start":"04:06.720 ","End":"04:08.505","Text":"It\u0027s like this,"},{"Start":"04:08.505 ","End":"04:14.655","Text":"not quite because it\u0027s not 1 and minus 1 it\u0027s Pi over 4 and minus Pi over 4,"},{"Start":"04:14.655 ","End":"04:18.625","Text":"and here it\u0027s 0,0, and 0."},{"Start":"04:18.625 ","End":"04:22.730","Text":"If we want to write this extension more formally,"},{"Start":"04:22.730 ","End":"04:24.259","Text":"this g Tilde,"},{"Start":"04:24.259 ","End":"04:27.785","Text":"it\u0027s Pi over 4 or minus Pi over 4 or 0."},{"Start":"04:27.785 ","End":"04:34.150","Text":"It will be Pi over 4 when x is between not just 0 and Pi,"},{"Start":"04:34.150 ","End":"04:38.690","Text":"but between any even number Pi and the following odd number Pi,"},{"Start":"04:38.690 ","End":"04:40.730","Text":"like between 4 Pi and 5 Pi,"},{"Start":"04:40.730 ","End":"04:43.205","Text":"or between 10 Pi and 11 Pi."},{"Start":"04:43.205 ","End":"04:45.455","Text":"If minus Pi over 4,"},{"Start":"04:45.455 ","End":"04:47.765","Text":"for example, between 3 Pi and 4 Pi,"},{"Start":"04:47.765 ","End":"04:50.195","Text":"between the odd and the following even,"},{"Start":"04:50.195 ","End":"04:52.898","Text":"and 0 at every multiple of Pi."},{"Start":"04:52.898 ","End":"04:55.580","Text":"I\u0027ll show you the picture again."},{"Start":"04:55.580 ","End":"05:00.110","Text":"It\u0027s this where this height is Pi over 4."},{"Start":"05:00.110 ","End":"05:02.245","Text":"This is minus Pi over 4."},{"Start":"05:02.245 ","End":"05:06.185","Text":"That actually answers the question that the series"},{"Start":"05:06.185 ","End":"05:10.670","Text":"converges to this function on all of the reals,"},{"Start":"05:10.670 ","End":"05:13.020","Text":"and there\u0027s its picture."}],"ID":28810},{"Watched":false,"Name":"Exercise 2 - Part c","Duration":"2m 10s","ChapterTopicVideoID":27566,"CourseChapterTopicPlaylistID":294459,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.790","Text":"Now we\u0027ve done Part A and B."},{"Start":"00:02.790 ","End":"00:04.755","Text":"We come to Part C,"},{"Start":"00:04.755 ","End":"00:10.065","Text":"where we have to compute the sum of this series of numbers,"},{"Start":"00:10.065 ","End":"00:12.990","Text":"which is, if you spell it out,"},{"Start":"00:12.990 ","End":"00:15.300","Text":"1 plus 1/3^2 plus 1/5,"},{"Start":"00:15.300 ","End":"00:17.700","Text":"plus 1/7^2 and so on to infinity."},{"Start":"00:17.700 ","End":"00:21.390","Text":"Now in Part B, we had this result,"},{"Start":"00:21.390 ","End":"00:24.090","Text":"at least on minus Pi to Pi."},{"Start":"00:24.090 ","End":"00:28.500","Text":"The sum of this series is given as follows,"},{"Start":"00:28.500 ","End":"00:30.495","Text":"and we named this g(x)."},{"Start":"00:30.495 ","End":"00:34.260","Text":"It actually converges to the periodic extension of g(x)."},{"Start":"00:34.260 ","End":"00:37.185","Text":"But we don\u0027t need that, this will do,"},{"Start":"00:37.185 ","End":"00:40.935","Text":"and what we have is,"},{"Start":"00:40.935 ","End":"00:44.280","Text":"we can write this as in Sigma form,"},{"Start":"00:44.280 ","End":"00:47.025","Text":"sum of 1/2n minus 1,"},{"Start":"00:47.025 ","End":"00:48.915","Text":"the sine of 2n minus 1x,"},{"Start":"00:48.915 ","End":"00:50.900","Text":"and that\u0027s equal to this function,"},{"Start":"00:50.900 ","End":"00:52.415","Text":"we\u0027ll call it g(x)."},{"Start":"00:52.415 ","End":"00:57.125","Text":"I want to apply Parseval\u0027s identity to this,"},{"Start":"00:57.125 ","End":"01:00.920","Text":"and we get 1/Pi times the integral of the absolute value"},{"Start":"01:00.920 ","End":"01:08.840","Text":"squared g(x) is equal to the sum of all the co-efficients and absolute value squared."},{"Start":"01:08.840 ","End":"01:12.650","Text":"In this case, we have the sum of absolute value of these squared,"},{"Start":"01:12.650 ","End":"01:16.880","Text":"is the integral of absolute value of this squared times 1/Pi."},{"Start":"01:16.880 ","End":"01:18.035","Text":"Let\u0027s do that."},{"Start":"01:18.035 ","End":"01:20.930","Text":"The right-hand side, we can drop the absolute value and put"},{"Start":"01:20.930 ","End":"01:24.225","Text":"the square in the denominator, and here,"},{"Start":"01:24.225 ","End":"01:28.915","Text":"the absolute value of g(x) is mostly Pi/4,"},{"Start":"01:28.915 ","End":"01:32.210","Text":"except for odd points which don\u0027t make any difference to the integral."},{"Start":"01:32.210 ","End":"01:35.850","Text":"The absolute value is plus Pi/4."},{"Start":"01:36.650 ","End":"01:41.555","Text":"It\u0027s Pi squared over 16 integral from minus Pi to Pi,"},{"Start":"01:41.555 ","End":"01:46.205","Text":"which means we multiply 2 Pi because it\u0027s a constant and we still have the 1/Pi,"},{"Start":"01:46.205 ","End":"01:53.385","Text":"and Pi squared times Pi/Pi is Pi squared and 2/16 is 1/8."},{"Start":"01:53.385 ","End":"01:56.985","Text":"So this converges to Pi squared over 8,"},{"Start":"01:56.985 ","End":"01:59.810","Text":"and if you want to spell it out without the Sigma,"},{"Start":"01:59.810 ","End":"02:06.060","Text":"you could write it as 4 terms and dot is Pi squared over 8,"},{"Start":"02:06.060 ","End":"02:08.190","Text":"and that concludes Part C,"},{"Start":"02:08.190 ","End":"02:10.570","Text":"and this is the last part."}],"ID":28811},{"Watched":false,"Name":"Exercise 3","Duration":"4m 47s","ChapterTopicVideoID":27567,"CourseChapterTopicPlaylistID":294459,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.734","Text":"In this exercise f is in this space,"},{"Start":"00:03.734 ","End":"00:08.114","Text":"sometimes called l2pc minus Pi Pi."},{"Start":"00:08.114 ","End":"00:12.390","Text":"Let c_n be its Fourier coefficients complex version,"},{"Start":"00:12.390 ","End":"00:16.935","Text":"and let d_n be the real part of c_n."},{"Start":"00:16.935 ","End":"00:23.085","Text":"Our task is to find the function f given that it satisfies the following 3 conditions."},{"Start":"00:23.085 ","End":"00:24.989","Text":"First of all, it\u0027s a real function."},{"Start":"00:24.989 ","End":"00:30.150","Text":"Secondly, it\u0027s 0 when x is between minus Pi and 0,"},{"Start":"00:30.150 ","End":"00:34.290","Text":"and thirdly given that the sum of"},{"Start":"00:34.290 ","End":"00:41.705","Text":"d_n e^inx is x^2 e^absolute value of x cosine x."},{"Start":"00:41.705 ","End":"00:43.675","Text":"Let\u0027s start solving it."},{"Start":"00:43.675 ","End":"00:46.670","Text":"C_n is the Fourier coefficient,"},{"Start":"00:46.670 ","End":"00:48.815","Text":"so let\u0027s interpret that."},{"Start":"00:48.815 ","End":"00:52.735","Text":"This is the formula for the Fourier coefficient, c_n."},{"Start":"00:52.735 ","End":"00:56.130","Text":"D_n is the real part of this,"},{"Start":"00:56.130 ","End":"01:00.120","Text":"so what we can get from this is that d_n is the real part of this."},{"Start":"01:00.120 ","End":"01:03.000","Text":"Now, 1 over 2Pi is real,"},{"Start":"01:03.000 ","End":"01:07.490","Text":"and you can push the real inside an integral,"},{"Start":"01:07.490 ","End":"01:09.140","Text":"and f(x) is real,"},{"Start":"01:09.140 ","End":"01:11.375","Text":"so it can push the real past that."},{"Start":"01:11.375 ","End":"01:16.945","Text":"We just have to take the real part of e^minus inx."},{"Start":"01:16.945 ","End":"01:26.005","Text":"This is cosine nx because e^minus inx is cosine nx minus i sine nx."},{"Start":"01:26.005 ","End":"01:30.725","Text":"Now remember f is 0 from minus Pi-0,"},{"Start":"01:30.725 ","End":"01:34.180","Text":"so we can change this integral to be just from 0-Pi."},{"Start":"01:34.180 ","End":"01:37.250","Text":"There\u0027s no point including the bits where it\u0027s 0,"},{"Start":"01:37.250 ","End":"01:38.980","Text":"so we get this."},{"Start":"01:38.980 ","End":"01:41.175","Text":"Now here\u0027s what we\u0027re going to do."},{"Start":"01:41.175 ","End":"01:49.035","Text":"We\u0027re going to let g be the even extension of f. I should\u0027ve said, minus Pi Pi."},{"Start":"01:49.035 ","End":"01:55.545","Text":"Now d_n; and I\u0027m just repeating here, is this."},{"Start":"01:55.545 ","End":"01:57.550","Text":"Now from 0-Pi,"},{"Start":"01:57.550 ","End":"02:00.080","Text":"f and g are the same,"},{"Start":"02:00.080 ","End":"02:03.990","Text":"so we can replace this f(x) by g(x)."},{"Start":"02:03.990 ","End":"02:05.820","Text":"I guess I skipped a step here."},{"Start":"02:05.820 ","End":"02:07.940","Text":"Then instead of 0-Pi,"},{"Start":"02:07.940 ","End":"02:11.195","Text":"we can put it from minus Pi-Pi and take half of it,"},{"Start":"02:11.195 ","End":"02:14.540","Text":"and that is because g is even. We have 2 steps."},{"Start":"02:14.540 ","End":"02:16.520","Text":"First, replace f with g,"},{"Start":"02:16.520 ","End":"02:21.380","Text":"and then extend to all of the symmetric interval where g is even."},{"Start":"02:21.380 ","End":"02:29.750","Text":"Next, I claim we can replace cosine nx by cosine nx minus i sine nx."},{"Start":"02:29.750 ","End":"02:34.720","Text":"We\u0027re going to be going backwards like here to get to e^minus inx."},{"Start":"02:34.720 ","End":"02:38.150","Text":"The reason we can do this; 2 integrals."},{"Start":"02:38.150 ","End":"02:42.290","Text":"If you think of it split as minus and then i times this integral"},{"Start":"02:42.290 ","End":"02:47.640","Text":"because g(x) sine nx is an odd function."},{"Start":"02:47.640 ","End":"02:49.730","Text":"This is even, this is odd."},{"Start":"02:49.730 ","End":"02:51.455","Text":"Product is odd."},{"Start":"02:51.455 ","End":"02:55.460","Text":"Integral and a symmetric interval is 0."},{"Start":"02:55.460 ","End":"02:59.884","Text":"We can add this bit because we\u0027re really adding or subtracting 0."},{"Start":"02:59.884 ","End":"03:10.530","Text":"Now we can replace cosine nx minus i sine nx with e^minus inx and we\u0027re almost done."},{"Start":"03:10.530 ","End":"03:12.465","Text":"This is what we have."},{"Start":"03:12.465 ","End":"03:14.260","Text":"If you look at this,"},{"Start":"03:14.260 ","End":"03:17.410","Text":"look at the half g(x),"},{"Start":"03:17.410 ","End":"03:19.555","Text":"that\u0027s a function,"},{"Start":"03:19.555 ","End":"03:24.040","Text":"and the rest of it is like the formula for the complex Fourier coefficient."},{"Start":"03:24.040 ","End":"03:26.440","Text":"1 over 2Pi integral of minus Pi-Pi,"},{"Start":"03:26.440 ","End":"03:29.350","Text":"and then times e^minus inx dx."},{"Start":"03:29.350 ","End":"03:33.805","Text":"This is the Fourier coefficient of 1/2 g(x),"},{"Start":"03:33.805 ","End":"03:39.532","Text":"which means that 1/2 g(x) has the following Fourier series,"},{"Start":"03:39.532 ","End":"03:42.230","Text":"sum of d_n e^inx."},{"Start":"03:42.230 ","End":"03:43.885","Text":"On the other hand,"},{"Start":"03:43.885 ","End":"03:50.650","Text":"we were told that x^2 e^absolute value of x cosine x was this same sum."},{"Start":"03:50.650 ","End":"03:55.830","Text":"These 2 must be equal and double it."},{"Start":"03:55.830 ","End":"03:58.470","Text":"This equals this. Twice this equals this,"},{"Start":"03:58.470 ","End":"04:00.015","Text":"so we have the 2 here."},{"Start":"04:00.015 ","End":"04:01.935","Text":"That\u0027s g(x)."},{"Start":"04:01.935 ","End":"04:03.885","Text":"What about f(x)?"},{"Start":"04:03.885 ","End":"04:10.759","Text":"Well, g(x) to remind you was the even extension of f, so it\u0027s this."},{"Start":"04:10.759 ","End":"04:17.855","Text":"Also remember that f is 0 when x is in the interval from minus Pi-0."},{"Start":"04:17.855 ","End":"04:22.595","Text":"On the positive part g=f or f=g let\u0027s say,"},{"Start":"04:22.595 ","End":"04:25.225","Text":"and on the negative part f is 0."},{"Start":"04:25.225 ","End":"04:31.055","Text":"F can be gotten by a combination of g when x is positive and 0 when x is negative,"},{"Start":"04:31.055 ","End":"04:35.100","Text":"so it\u0027s equal to this when x is"},{"Start":"04:35.100 ","End":"04:39.980","Text":"positive and 0 when x is negative all in the interval from minus Pi-Pi."},{"Start":"04:39.980 ","End":"04:42.185","Text":"This is what f(x) is,"},{"Start":"04:42.185 ","End":"04:47.340","Text":"and that\u0027s what we were asked for and we are done."}],"ID":28812},{"Watched":false,"Name":"Exercise 4","Duration":"8m 1s","ChapterTopicVideoID":27568,"CourseChapterTopicPlaylistID":294459,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.400","Text":"In this exercise, f is a 2Pi-periodic even function such that f(x) is cosine 2x"},{"Start":"00:08.400 ","End":"00:16.995","Text":"for x between 0 and Pi over 2 and f(x) is minus 1 for x between Pi over 2 and Pi."},{"Start":"00:16.995 ","End":"00:19.395","Text":"Let me just explain this."},{"Start":"00:19.395 ","End":"00:26.190","Text":"This is the part where we\u0027ve defined it at just from 0 to Pi."},{"Start":"00:26.190 ","End":"00:29.190","Text":"We can make it over all of r by doing two things."},{"Start":"00:29.190 ","End":"00:32.820","Text":"First of all, take a reflection of this about the y-axis."},{"Start":"00:32.820 ","End":"00:38.160","Text":"We get this part up to minus Pi and then repeat,"},{"Start":"00:38.160 ","End":"00:42.965","Text":"copy paste, all the way to plus and minus infinity."},{"Start":"00:42.965 ","End":"00:45.875","Text":"The even part doubles it,"},{"Start":"00:45.875 ","End":"00:50.370","Text":"and the 2Pi-periodic makes it continue to all of the reals."},{"Start":"00:50.370 ","End":"00:51.855","Text":"This is the graph."},{"Start":"00:51.855 ","End":"01:00.840","Text":"Now we have to find the real Fourier series of f and then compute the sum as follows."},{"Start":"01:00.840 ","End":"01:04.940","Text":"Then there\u0027s the question, does the Fourier series of f converge to it uniformly?"},{"Start":"01:04.940 ","End":"01:08.750","Text":"Explain. There\u0027s really three parts to this question."},{"Start":"01:08.750 ","End":"01:10.130","Text":"I start solving it."},{"Start":"01:10.130 ","End":"01:16.615","Text":"Now, the even part tells us that we can use the shortcut formula for even functions."},{"Start":"01:16.615 ","End":"01:18.870","Text":"We don\u0027t have any b_ns,"},{"Start":"01:18.870 ","End":"01:21.885","Text":"and we just have this,"},{"Start":"01:21.885 ","End":"01:26.345","Text":"and a_n is given as the integral instead of a minus Pi to Pi,"},{"Start":"01:26.345 ","End":"01:28.820","Text":"just from 0 to Pi, but there\u0027s 2 here."},{"Start":"01:28.820 ","End":"01:31.595","Text":"Now we can break this integral up into 2 parts."},{"Start":"01:31.595 ","End":"01:39.130","Text":"The part from 0 to Pi over 2 and the part from Pi over 2 to Pi, like so."},{"Start":"01:39.130 ","End":"01:43.830","Text":"Now, from 0 to Pi over 2 f(x) is cosine 2x,"},{"Start":"01:43.830 ","End":"01:46.510","Text":"and the other part f(x) is minus 1."},{"Start":"01:46.510 ","End":"01:49.810","Text":"We just make it a minus here and there\u0027s 1 here."},{"Start":"01:49.810 ","End":"01:52.230","Text":"I\u0027ll give you the answers for the integrals,"},{"Start":"01:52.230 ","End":"01:54.245","Text":"we don\u0027t want to waste too much time with integration."},{"Start":"01:54.245 ","End":"02:00.860","Text":"The integral of cosine 2x cosine nx from 0 to Pi over 2 comes out to be this,"},{"Start":"02:00.860 ","End":"02:03.940","Text":"this part is this and this part is this."},{"Start":"02:03.940 ","End":"02:06.830","Text":"If you need the computations, I wrote them out."},{"Start":"02:06.830 ","End":"02:08.810","Text":"You can pause and take a look at them,"},{"Start":"02:08.810 ","End":"02:11.095","Text":"didn\u0027t want to waste time with it."},{"Start":"02:11.095 ","End":"02:13.095","Text":"We can simplify this."},{"Start":"02:13.095 ","End":"02:16.800","Text":"The sine nPi over 2 we can take out the brackets,"},{"Start":"02:16.800 ","End":"02:23.160","Text":"and we\u0027re left with n over 4 minus n^2 plus 1 over n. We can do an addition here."},{"Start":"02:23.160 ","End":"02:26.790","Text":"We will get n^2 plus 4 minus n^2,"},{"Start":"02:26.790 ","End":"02:29.795","Text":"which is just 4 and this times this."},{"Start":"02:29.795 ","End":"02:35.060","Text":"I forgot to say this only works if n is not equal to 0 or 2,"},{"Start":"02:35.060 ","End":"02:37.610","Text":"because we don\u0027t want these denominators to be 0."},{"Start":"02:37.610 ","End":"02:41.785","Text":"We\u0027ll have to work out a_naught and a_2 separately."},{"Start":"02:41.785 ","End":"02:46.185","Text":"A_naught, if you go back and look at the formula,"},{"Start":"02:46.185 ","End":"02:49.920","Text":"it\u0027s cosine 2x, cosine nx."},{"Start":"02:49.920 ","End":"02:51.240","Text":"It\u0027s cosine 0x,"},{"Start":"02:51.240 ","End":"02:55.360","Text":"so it\u0027s just cosine 2x here and here it\u0027s just 1."},{"Start":"02:55.360 ","End":"02:58.730","Text":"This integral comes out to be 0."},{"Start":"02:58.730 ","End":"03:03.080","Text":"This integral is just the length of the interval which is Pi over 2,"},{"Start":"03:03.080 ","End":"03:05.430","Text":"comes out to be minus 1,"},{"Start":"03:05.430 ","End":"03:09.685","Text":"2 over Pi times Pi over 2 minus is minus 1."},{"Start":"03:09.685 ","End":"03:12.180","Text":"Now a_2, cosine 2x,"},{"Start":"03:12.180 ","End":"03:18.730","Text":"cosine 2x is cosine^2 2x and here minus integral of cosine (2x) dx."},{"Start":"03:18.730 ","End":"03:22.220","Text":"This tell you the answer comes out to be Pi over 4."},{"Start":"03:22.220 ","End":"03:24.050","Text":"This, just like here,"},{"Start":"03:24.050 ","End":"03:28.300","Text":"is 0, 2 over Pi times Pi over 4 is 1/2."},{"Start":"03:28.300 ","End":"03:30.170","Text":"Now we can combine this,"},{"Start":"03:30.170 ","End":"03:32.180","Text":"f(x) is the following formula."},{"Start":"03:32.180 ","End":"03:35.435","Text":"Just plug in a_naught and a_0 and a_2."},{"Start":"03:35.435 ","End":"03:39.785","Text":"We get a_naught over 2 is minus 1/2,"},{"Start":"03:39.785 ","End":"03:47.910","Text":"a_2 times cosine 2x is this and the sum just remove the n equals 2-part."},{"Start":"03:47.910 ","End":"03:50.055","Text":"Just indicate it like so."},{"Start":"03:50.055 ","End":"03:53.230","Text":"Here again is the picture."},{"Start":"03:53.230 ","End":"03:58.340","Text":"I claim that f is continuous at x"},{"Start":"03:58.340 ","End":"04:03.560","Text":"equals naught because we mirror imaged it because it\u0027s an even function."},{"Start":"04:03.560 ","End":"04:08.630","Text":"Whatever it is, when you make a mirror image is going to be continuous at 0."},{"Start":"04:08.630 ","End":"04:14.975","Text":"Also, the 1 sided derivatives exist at 0."},{"Start":"04:14.975 ","End":"04:19.395","Text":"Here the 1 side derivative is 0 and here it\u0027s 0,"},{"Start":"04:19.395 ","End":"04:22.020","Text":"in fact they are equal even as a derivative at 0,"},{"Start":"04:22.020 ","End":"04:24.555","Text":"but we just need 1-sided derivatives."},{"Start":"04:24.555 ","End":"04:29.585","Text":"That\u0027s for Dirichlet\u0027s theorem that requires this because it satisfies these,"},{"Start":"04:29.585 ","End":"04:34.070","Text":"we can substitute x equals 0 and instead of this wave,"},{"Start":"04:34.070 ","End":"04:35.935","Text":"this tilt, we can put equals."},{"Start":"04:35.935 ","End":"04:38.630","Text":"We get f(0) is the following;"},{"Start":"04:38.630 ","End":"04:41.150","Text":"cosine twice 0 is cosine 0 is 1,"},{"Start":"04:41.150 ","End":"04:42.650","Text":"so get rid of that."},{"Start":"04:42.650 ","End":"04:44.870","Text":"Cosine 0 is 1 again,"},{"Start":"04:44.870 ","End":"04:47.420","Text":"and minus 1/2 plus 1/2 is 0."},{"Start":"04:47.420 ","End":"04:55.425","Text":"We just have this part and f(0) is 1 because it\u0027s cosine twice 0, cosine 2x."},{"Start":"04:55.425 ","End":"04:57.410","Text":"When n is even,"},{"Start":"04:57.410 ","End":"05:01.610","Text":"sine of nPi over 2 is going to be 0."},{"Start":"05:01.610 ","End":"05:07.235","Text":"Because n over 2 will be a whole number and the whole number times Pi has a sine of 0."},{"Start":"05:07.235 ","End":"05:09.770","Text":"We just need to take odd numbers for n,"},{"Start":"05:09.770 ","End":"05:11.180","Text":"we can say 2k plus 1,"},{"Start":"05:11.180 ","End":"05:13.190","Text":"where k is 0,1, 2, etc."},{"Start":"05:13.190 ","End":"05:16.115","Text":"We get that 1 is equal 2."},{"Start":"05:16.115 ","End":"05:22.260","Text":"Instead of this, we can put 2k plus 1 instead of n. Here,"},{"Start":"05:22.260 ","End":"05:23.550","Text":"n is 2k plus 1,"},{"Start":"05:23.550 ","End":"05:25.830","Text":"2 minus n is 1 minus 2k,"},{"Start":"05:25.830 ","End":"05:29.130","Text":"and 2 plus n is 2k plus 3."},{"Start":"05:29.130 ","End":"05:32.055","Text":"Bring the Pi over 2 to the other side,"},{"Start":"05:32.055 ","End":"05:34.320","Text":"and then we have this,"},{"Start":"05:34.320 ","End":"05:41.140","Text":"sine of 2k plus 1 over 2 Pi is sine of kPi plus Pi over 2."},{"Start":"05:41.140 ","End":"05:42.890","Text":"We\u0027ve seen this before,"},{"Start":"05:42.890 ","End":"05:47.790","Text":"sine of a multiple of Pi plus Pi over 2 is"},{"Start":"05:47.790 ","End":"05:52.910","Text":"minus 1 to this k. Here\u0027s the work if you need it, we\u0027ve seen it before."},{"Start":"05:52.910 ","End":"05:59.240","Text":"I\u0027m replacing this by minus 1 to the k. Also bring the minus 4 over to the other side,"},{"Start":"05:59.240 ","End":"06:01.690","Text":"so it\u0027s minus Pi over 8."},{"Start":"06:01.690 ","End":"06:07.740","Text":"Continuing. What we\u0027re going to do is break this sum up into 2 pieces,"},{"Start":"06:07.740 ","End":"06:09.790","Text":"just the first term where k is 0 and"},{"Start":"06:09.790 ","End":"06:13.195","Text":"then all the rest of it because we were asked for the sum,"},{"Start":"06:13.195 ","End":"06:15.625","Text":"k goes from 1 to infinity."},{"Start":"06:15.625 ","End":"06:18.790","Text":"When k is 0, we have here 1,"},{"Start":"06:18.790 ","End":"06:21.700","Text":"here 1 minus 1 and 3,"},{"Start":"06:21.700 ","End":"06:24.790","Text":"and then the sum from 1 to infinity."},{"Start":"06:24.790 ","End":"06:27.595","Text":"This is equal to minus 1/3."},{"Start":"06:27.595 ","End":"06:32.440","Text":"Then we can bring this to this side or bring this to this side."},{"Start":"06:32.440 ","End":"06:33.850","Text":"I need one other thing,"},{"Start":"06:33.850 ","End":"06:35.230","Text":"replace k with n,"},{"Start":"06:35.230 ","End":"06:37.420","Text":"it doesn\u0027t matter what variable we use."},{"Start":"06:37.420 ","End":"06:39.845","Text":"But this is because we were asked for this sum,"},{"Start":"06:39.845 ","End":"06:44.235","Text":"and so the answer to that part is 1/3 minus Pi over 8."},{"Start":"06:44.235 ","End":"06:46.940","Text":"There\u0027s one question we still haven\u0027t answered."},{"Start":"06:46.940 ","End":"06:50.360","Text":"Does the Fourier series of f converge to f uniformly?"},{"Start":"06:50.360 ","End":"06:51.845","Text":"The answer is yes."},{"Start":"06:51.845 ","End":"06:54.920","Text":"We\u0027ll use the theorem on uniform convergence."},{"Start":"06:54.920 ","End":"06:58.370","Text":"If f is continuous on minus Pi,"},{"Start":"06:58.370 ","End":"07:07.190","Text":"Pi and equal at the endpoints and f\u0027 is piecewise continuous on minus Pi,"},{"Start":"07:07.190 ","End":"07:11.840","Text":"Pi, then the Fourier series does converge uniformly."},{"Start":"07:11.840 ","End":"07:14.400","Text":"F is continuous,"},{"Start":"07:14.400 ","End":"07:16.835","Text":"only 3 possible problems."},{"Start":"07:16.835 ","End":"07:19.580","Text":"Here it\u0027s continuous because of evenness."},{"Start":"07:19.580 ","End":"07:22.478","Text":"Here it\u0027s continuous again because of evenness."},{"Start":"07:22.478 ","End":"07:25.780","Text":"Then here because the way it was defined,"},{"Start":"07:25.780 ","End":"07:28.740","Text":"cosine of 2x here is minus 1,"},{"Start":"07:28.740 ","End":"07:35.059","Text":"the same as this and f\u0027 is piecewise continuous."},{"Start":"07:35.059 ","End":"07:39.885","Text":"It\u0027s continuous from 1 to Pi over 2 open interval."},{"Start":"07:39.885 ","End":"07:41.550","Text":"From here to here,"},{"Start":"07:41.550 ","End":"07:44.675","Text":"it doesn\u0027t matter what happens at this point exactly."},{"Start":"07:44.675 ","End":"07:47.720","Text":"Although it turns out that it\u0027s also continuous because"},{"Start":"07:47.720 ","End":"07:51.260","Text":"the derivative of f\u0027 is 0 on the left and on the right,"},{"Start":"07:51.260 ","End":"07:53.720","Text":"and here it\u0027s also 0 on the left and on the right."},{"Start":"07:53.720 ","End":"07:56.275","Text":"Anyway, all the conditions are met."},{"Start":"07:56.275 ","End":"07:58.965","Text":"It does converge uniformly."},{"Start":"07:58.965 ","End":"08:01.900","Text":"That concludes this exercise."}],"ID":28813},{"Watched":false,"Name":"Exercise 5","Duration":"5m 25s","ChapterTopicVideoID":27569,"CourseChapterTopicPlaylistID":294459,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.805","Text":"In this exercise, we take a general function f,"},{"Start":"00:03.805 ","End":"00:08.180","Text":"it\u0027s piecewise continuous and 2Pi periodic."},{"Start":"00:08.180 ","End":"00:12.545","Text":"Let\u0027s say its Fourier series is the sum of"},{"Start":"00:12.545 ","End":"00:20.395","Text":"f_n e^inx complex Fourier series and let h be some positive number."},{"Start":"00:20.395 ","End":"00:25.150","Text":"We define a new function g(x) in terms of f as follows;"},{"Start":"00:25.150 ","End":"00:28.285","Text":"1/2h, the integral from minus h to h,"},{"Start":"00:28.285 ","End":"00:35.040","Text":"f(x) plus tdt is an integral with respect to t and so in the end,"},{"Start":"00:35.040 ","End":"00:36.130","Text":"t drops out,"},{"Start":"00:36.130 ","End":"00:38.395","Text":"and we have a function of x."},{"Start":"00:38.395 ","End":"00:44.030","Text":"Our task is to find the Fourier coefficients g_n of g in"},{"Start":"00:44.030 ","End":"00:49.160","Text":"terms of the Fourier coefficients f_n of f. By the way,"},{"Start":"00:49.160 ","End":"00:57.320","Text":"this function g could be described as the average on the range from x minus h to x"},{"Start":"00:57.320 ","End":"01:06.520","Text":"plus h of f. It\u0027s a bit like f(x) but spread out on an interval of length 2h."},{"Start":"01:06.520 ","End":"01:12.365","Text":"Let\u0027s start with noting that f_n is given by the standard formula,"},{"Start":"01:12.365 ","End":"01:14.704","Text":"just recording this now for later."},{"Start":"01:14.704 ","End":"01:20.415","Text":"Now let\u0027s start with g. Standard formula for the Fourier series g is the sum of"},{"Start":"01:20.415 ","End":"01:27.110","Text":"g_n e^inx and g_n is given by this integral with a minus here."},{"Start":"01:27.110 ","End":"01:32.090","Text":"Now recall that g(x) just taken from here is the following,"},{"Start":"01:32.090 ","End":"01:36.050","Text":"so we can substitute that in here and get the following,"},{"Start":"01:36.050 ","End":"01:37.820","Text":"an integral of an integral."},{"Start":"01:37.820 ","End":"01:40.940","Text":"We can write this as a double integral."},{"Start":"01:40.940 ","End":"01:43.610","Text":"Just take the constant out in front."},{"Start":"01:43.610 ","End":"01:46.085","Text":"We have a double integral, dxdt."},{"Start":"01:46.085 ","End":"01:48.620","Text":"When you switch the order of the integrals,"},{"Start":"01:48.620 ","End":"01:51.710","Text":"we can make the integral with respect to x,"},{"Start":"01:51.710 ","End":"01:54.260","Text":"the inner one, and dt the outer one."},{"Start":"01:54.260 ","End":"01:59.900","Text":"We\u0027re trying to get this to be expressed in terms of f_n, so we need f(x),"},{"Start":"01:59.900 ","End":"02:01.250","Text":"not f(x) plus t,"},{"Start":"02:01.250 ","End":"02:06.380","Text":"so next step will be to do a substitution."},{"Start":"02:06.380 ","End":"02:11.210","Text":"Let u equal x plus t and then du equals dx and x"},{"Start":"02:11.210 ","End":"02:16.250","Text":"is u minus t. What we get is the following,"},{"Start":"02:16.250 ","End":"02:21.965","Text":"where the limits of integration are minus 2Pi plus t,"},{"Start":"02:21.965 ","End":"02:27.380","Text":"and this x is u minus t and x plus t is u."},{"Start":"02:27.380 ","End":"02:29.030","Text":"If we multiply this out,"},{"Start":"02:29.030 ","End":"02:33.505","Text":"it\u0027s e to the minus inu each of the plus int,"},{"Start":"02:33.505 ","End":"02:40.635","Text":"and the e to the int can be brought in front because this is integral du,"},{"Start":"02:40.635 ","End":"02:44.220","Text":"and e^int is not dependent on u."},{"Start":"02:44.220 ","End":"02:48.210","Text":"This function here is 2Pi periodic."},{"Start":"02:48.210 ","End":"02:53.300","Text":"F is 2Pi periodic and e to the minus inu is 2Pi periodic."},{"Start":"02:53.300 ","End":"02:56.105","Text":"If you replace u by u plus 2Pi,"},{"Start":"02:56.105 ","End":"02:59.570","Text":"you get same thing as a property of"},{"Start":"02:59.570 ","End":"03:01.910","Text":"2Pi periodic functions that if you take the"},{"Start":"03:01.910 ","End":"03:05.300","Text":"integral over any interval of the same length,"},{"Start":"03:05.300 ","End":"03:08.150","Text":"in this case, 2Pi, you\u0027ll get the same thing."},{"Start":"03:08.150 ","End":"03:12.935","Text":"The integral from here to here would be the same as if we shifted"},{"Start":"03:12.935 ","End":"03:18.865","Text":"along by a quantity A because whatever lost here is gained here."},{"Start":"03:18.865 ","End":"03:21.185","Text":"Similarly, in our case,"},{"Start":"03:21.185 ","End":"03:24.050","Text":"this is an interval of length 2Pi,"},{"Start":"03:24.050 ","End":"03:28.270","Text":"and we can just replace it by minus Pi to Pi."},{"Start":"03:28.270 ","End":"03:30.880","Text":"Everything else untouched."},{"Start":"03:30.880 ","End":"03:32.860","Text":"It\u0027s like here it was sliding it along."},{"Start":"03:32.860 ","End":"03:36.455","Text":"We can bring the 1/2Pi inside here,"},{"Start":"03:36.455 ","End":"03:43.285","Text":"and we get this and this is just like what we had earlier, I\u0027ll scroll back."},{"Start":"03:43.285 ","End":"03:46.580","Text":"Yeah, this is f_n."},{"Start":"03:46.580 ","End":"03:49.060","Text":"Now we have it in terms of u instead of x,"},{"Start":"03:49.060 ","End":"03:50.290","Text":"but that makes no difference,"},{"Start":"03:50.290 ","End":"03:52.825","Text":"the variable is a dummy."},{"Start":"03:52.825 ","End":"03:55.750","Text":"This is the following."},{"Start":"03:55.750 ","End":"03:58.760","Text":"Just replacing this by f_n."},{"Start":"03:58.760 ","End":"04:02.265","Text":"This is f_n."},{"Start":"04:02.265 ","End":"04:05.080","Text":"You can bring this out in front because it\u0027s a constant."},{"Start":"04:05.080 ","End":"04:07.510","Text":"At least it doesn\u0027t depend on t, depends on n,"},{"Start":"04:07.510 ","End":"04:11.150","Text":"but not on t, bringing it out in front, and we have this."},{"Start":"04:11.150 ","End":"04:17.225","Text":"Continuing, we could write e^int in terms of real and imaginary parts."},{"Start":"04:17.225 ","End":"04:20.360","Text":"It\u0027s cosine nt plus i sine nt,"},{"Start":"04:20.360 ","End":"04:22.340","Text":"and then we have 2 integrals,"},{"Start":"04:22.340 ","End":"04:24.950","Text":"real one and an imaginary one."},{"Start":"04:24.950 ","End":"04:28.325","Text":"Now, this is an odd function,"},{"Start":"04:28.325 ","End":"04:30.740","Text":"and this is an even function."},{"Start":"04:30.740 ","End":"04:34.020","Text":"An odd function on a symmetric interval has an integral of zeros,"},{"Start":"04:34.020 ","End":"04:35.660","Text":"so this drops out."},{"Start":"04:35.660 ","End":"04:41.135","Text":"This one we could write as twice the integral from 0 to h,"},{"Start":"04:41.135 ","End":"04:42.755","Text":"so this is what we get."},{"Start":"04:42.755 ","End":"04:46.700","Text":"Now this integral is equal to sine nt,"},{"Start":"04:46.700 ","End":"04:48.925","Text":"providing n is not 0."},{"Start":"04:48.925 ","End":"04:52.350","Text":"G_n, if n is not 0,"},{"Start":"04:52.350 ","End":"04:55.875","Text":"is sine nh over nh,"},{"Start":"04:55.875 ","End":"04:58.590","Text":"when you plug in 0, you get 0."},{"Start":"04:58.590 ","End":"05:02.530","Text":"It\u0027s just what happens when you plug in h, that\u0027s g_n,"},{"Start":"05:02.530 ","End":"05:04.780","Text":"and we want also g naught,"},{"Start":"05:04.780 ","End":"05:10.290","Text":"so put 0 instead of n. Now this is 1."},{"Start":"05:10.290 ","End":"05:12.810","Text":"The integral of 1 from 0 to h is h,"},{"Start":"05:12.810 ","End":"05:14.625","Text":"h over h is 1,"},{"Start":"05:14.625 ","End":"05:16.715","Text":"so this is just f naught."},{"Start":"05:16.715 ","End":"05:21.380","Text":"This together with this gives the formula for all the g_n in"},{"Start":"05:21.380 ","End":"05:26.680","Text":"terms of the f_n and that\u0027s the end of this exercise."}],"ID":28814}],"Thumbnail":null,"ID":294459}]

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1.1

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