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

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

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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 < ,- 1 , < 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)...
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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.

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

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