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Unformatted text preview: R. Balan Homework #10 Solutions MATH 464 I . (a) Poisson's summation formula implies: f ( x ) = X m = e ( x mp ) 2 = 1 p X k = e 2 ikx/p F ( k/p ) where F ( s ) = F ( e x 2 )( s ) = e s 2 Thus: f ( x ) = 1 p X k = e 2 ikx/p e k 2 /p 2 which is the Fourier series expansion of f ( x ) . (b) Poisson's sum formula implies: f ( x ) = X m = e a  x mp  = 1 p X k = e 2 ikx/p F ( k/p ) where F ( s ) = F ( e a  x  )( s ) = 2 a a 2 + 4 2 s 2 Thus f ( x ) = 1 p X k = e 2 ikx/p 2 ap 2 a 2 p 2 + 4 2 k 2 which is the Fourier series expansion of f ( x ) . II . Recall c n = R 1 e 2 inx f ( x ) dx . Thus (a) Assume f ( x ) = f ( x ) . Thus: c n = Z 1 e 2 inx f ( x ) dx = Z 1 e 2 inx f ( x ) dx = c n . (b) Assume f ( x ) = f ( x ) . Thus: c n = Z 1 e 2 inx f ( x ) dx = Z 1 e 2 inx f ( x ) dx = c n . (c) Assume f ( x + 1 /m ) = f ( x ) . Thus: c n = m 1 X k =0 Z ( k +1) /m k/m e 2 inx f ( x ) dx = m 1 X...
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This note was uploaded on 05/05/2010 for the course MATH 464 taught by Professor Staff during the Spring '08 term at Maryland.
 Spring '08
 staff
 Fourier Series

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