Sol-HW5 - Math Methods Solutions for Assignment 5 Oct. 04,...

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Unformatted text preview: Math Methods Solutions for Assignment 5 Oct. 04, 2010 Fall 2010 Fourier Series and Transforms Due Oct. 11, 2010 1. Prove Parsevals Theorem. If f ( x ) a 2 + X n =1 ( a n cos( nx ) + b n sin( nx )) then Z - f 2 ( x ) dx = 2 a 2 + X n =1 ( a 2 n + b 2 n ) Z - f 2 ( x ) dx = Z - " a 2 4 + X n =1 X m =1 ( a n a m cos( nx )cos( mx ) + b n b m sin( nx )sin( mx )+ + a n b m cos( nx )sin( mx ) + a m b n cos( mx )sin( nx )] dx = 2 a 2 + X n =1 X m =1 ( a n a m nm + b n b m nm ) 2. Show that for even symmetric function f ( 2 + x ) = f ( 2- x ), Fourier series has b n = 0 and a 2 n +1 = 0. For even function b n = 0. a n = 1 Z - f ( x )cos nxdx = 1 Z / 2- 3 / 2 f 2 + t cos h n 2 + t i dt , where we made a change of variables: x = 2 + t . Due to symmetry a n = 1 Z / 2- 3 / 2 f 2- t cos h n 2 + t i dt = 1 Z 2 f ( y )cos[ n ( - y )] dy , where y = 2- t ....
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Sol-HW5 - Math Methods Solutions for Assignment 5 Oct. 04,...

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