X1 to calculate the rate of change of heat within dx1

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: , extended to be periodic of period 2π . b. Use (3.19) to find the real form of the Fourier series. 3.4.4. Show that if a function f : R → R is smooth and periodic of period 2πL, we can write the Fourier expansion of f as f (t) = . . . + c−2 e−2it/L + c−1 e−it/L + c0 + c1 eit/L + c2 e2it/L + . . . , where ck = 3.5 1 2πL πL f (t)e−ikt/L dt. −πL Fourier transforms* One of the problems with the theory of Fourier series presented in the previous sections is that it applies only to periodic functions. There are many times when one would like to divide a function which is not periodic into a superposition of sines and cosines. The Fourier transform is the tool often used for this purpose. The idea behind the Fourier transform is to think of f (t) as vanishing outside a very long interval [−πL, πL]. The function can be extended to a periodic function f (t) such that f (t + 2πL) = f (t). According to the theory of Fourier series in complex form (see Exercise 3.4.4), f (t) = . . . + c−2 e−2it/L + c−1 e−it/L + c0 + c1 eit/L + c2 e2it/L + . . . , where the ck ’s are the complex numbers. De...
View Full Document

This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online