Fourier-x

Fourier-x - Fourier Series of f (x) = x To write: eikx . x...

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

View Full Document Right Arrow Icon
Fourier Series of f ( x ) = x To write: x = k c k e ikx 2 π . The Fourier coefficients are c k = 1 2 π Z π - π xe - ikx dx = 1 2 π Z π - π x [ cos kx - i sin kx ] dx = - 2 i 2 π Z π 0 x sin kxdx But Z π 0 x sin kxdx = - x cos kx k π 0 + 1 k Z π 0 cos kxdx = - π cos k π k = - π k ( - 1 ) k . Thus c k = - 2 i 2 π h - π k ( - 1 ) k i = i 2 π ± ( - 1 ) k k ² . Consequently x = i 2 π k 6 = 0 ( - 1 ) k k e ikx 2 π = i k 6 = 0 ( - 1 ) k k e ikx = - 2 k = 1 ( - 1 ) k k sin kx = 2 ± sin x - sin2 x 2 + sin3 x 3 - sin4 x 4 + ··· ² Finally we compute k x k 2 = | c k | 2 . Since k x k
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 03/06/2012.

Ask a homework question - tutors are online