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Sol-HW5 - Math Methods Fall 2010 Solutions for Assignment 5...

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Math Methods Solutions for Assignment 5 Oct. 04, 2010 Fall 2010 Fourier Series and Transforms Due Oct. 11, 2010 1. Prove Parseval’s Theorem. If f ( x ) a 0 2 + X n =1 ( a n cos( nx ) + b n sin( nx )) then Z π - π f 2 ( x ) dx = π 2 a 2 0 + π X n =1 ( a 2 n + b 2 n ) Z π - π f 2 ( x ) dx = Z π - π " a 2 0 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 = π 2 a 2 0 + π X n =1 X m =1 ( a n a m δ nm + b n b m δ nm ) 2. Show that for even symmetric function f ( π 2 + x ) = f ( π 2 - x ), Fourier series has b n = 0 and a 2 n +1 = 0. For even function b n = 0. a n = 1 π Z π - π f ( x ) cos nxdx = 1 π Z π/ 2 - 3 π/ 2 f π 2 + t cos h n π 2 + t i dt , where we made a change of variables: x = π 2 + t . Due to symmetry a n = 1 π Z π/ 2 - 3 π/ 2 f π 2 - t cos h n π 2 + t i dt = 1 π Z 2 π 0 f ( y ) cos[ n ( π - y )] dy , where y = π 2 - t . Hence a n = 1 π Z 2 π 0 f ( y )( - 1) n cos( ny ) dy = ( - 1) n a n , i.e. a n = 0 for odd n . 3. Consider the step function f ( x ) = π for 0 x < π - π for - π x < 0 Find a) the Fourier series for f ( x ),
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b) integral of Fourier series, c) show that differential of the Fourier series does not exist.
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