9HE.Solutions.to.Special.Probs.Chaps.5-6

# 9HE.Solutions.to.Special.Probs.Chaps.5-6 - Chapter 5:...

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

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

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

Unformatted text preview: Chapter 5: Problem S1 A periodic function f ( x ) can be written in terms of sines and cosines as f ( x ) = a 2 + X n =1 [ a n cos( nx ) + b n sin( nx )] (1) where the Fourier coefficients a n and b n are given by a n = 1 Z - f ( x ) cos( nx ) dx, b n = 1 Z - f ( x ) sin( nx ) dx. (2) We can write the expression for the square wave as f ( x ) = 1 ,- x &lt; ,- 1 , &lt; x , (3) so the a n coefficients are a n = 1 Z- cos( nx ) dx- 1 Z cos( nx ) dx = 0 . (4) Because cosine is an even function, the two terms cancel. That the a n are zero should be obvious without doing the work since our f ( x ) is odd. The b n are b n = 1 Z- sin( nx ) dx- 1 Z sin( nx ) dx =- 2 Z sin( nx ) dx = 2 n cos( nx ) . (5) When n is even, cos( n ) is one, as is cos(0) and the two cancel. When n is even, the two terms add. As a result, the first two non-zero coefficients are given by n = 1 and n = 3: b 1 =- 4 - 1 . 27 , b 3 =- 4 3 - . 42 (6)...
View Full Document

## This note was uploaded on 03/01/2012 for the course PHY 9HE taught by Professor Charlesfadley during the Winter '11 term at UC Davis.

### Page1 / 3

9HE.Solutions.to.Special.Probs.Chaps.5-6 - Chapter 5:...

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

View Full Document
Ask a homework question - tutors are online