c22 - 18.03 Class 22, April 5 Fourier series and harmonic...

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18.03 Class 22, April 5 Fourier series and harmonic response [1] My muddy point from the last lecture: I claimed that the Fourier series for f(t) converges wherever $f$ is continuous. What does this really say? For example, (pi/4) sq(t) = sin(t) + (1/3) sin(3t) + (1/5) sin(5t) + . ... for any value of t which is not a whole number multiple of pi. This says that when 0 < t < pi , sin(t) + (1/3) sin(3t) + . ... = pi/4 (*) Since all these sines are odd functions, it is no additional information to say that when -pi < t < 0 , sin(t) + (1/3) sin(3t) + . ... = - pi/4 For example we could take t = pi/2 . Then sin(pi/2) = 1 , sin(3pi/2) = -1 , sin(5pi/2) = 1 , .... so 1 - (1/3) + (1/5) - (1/7) + . .. = pi/4 This is an alternating series, so we know it converges. Did you know that it converges to pi/4? And so on: there are infinitely many summations like this contained in (*) . [2] You can differentiate and integrate Fourier series. Example: Consider the function f(t) which is periodic of period 2pi and is given by f(t) = (pi/2) - t between 0 and pi. We could calculate the coefficients, using the fact that f(t) is even and integration by parts. For a start, a0/2 is the average value, which is pi/2. Or we could realize that
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c22 - 18.03 Class 22, April 5 Fourier series and harmonic...

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