MIT18_03S10_c22

MIT18_03S10_c22 - 18.03 Class 22 Fourier Series III[1...

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18.03 Class 22 , March 31, 2010 Fourier Series III [1] Differentiation and integration [2] Harmonic oscillator with periodic input [3] What you hear [1] 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) = |t| between -pi 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 f'(t) = sq(t) (except where f'(t) doesn't exist) or what is the same f(t) = integral_0^t sq(u) du and integrate the Fourier series of the squarewave. NB: it is not true in general that the integral of a periodic function is periodic; think of integrating the constant function 1 for example. But the integral IS periodic if the average value of the function is zero. If you think of this one term at a time, the point is that the integral of cos(nt) is periodic unless n = 0 and the integral of sin(nt) is always periodic. Let's compute: f(t) = integral_0^t (4/pi) sum_{n odd} sin(nx)/n dx = (4/pi) sum_{n odd} integral_0^t sin(nx)/n dx = (4/pi) sum_{n odd} [- cos(nx) / n^2]_0^t
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