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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 = (4/pi) sum_{n odd} (1/n^2) - (4/pi) sum_{n odd} cos(nt) / n^2 That's it, that's the Fourier series for...
View Full Document

This note was uploaded on 10/21/2011 for the course MATH 18.03 taught by Professor Vogan during the Spring '09 term at MIT.

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online