Homework3_11

# Homework3_11 - otherwise The Fourier coeﬃcient c j of the...

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Math Biophysics, Fall 2011 Homework 3 (1) Recall the delta function for the integer lattice δ Z can be approximated by N X k = - N e 2 πikx . Show the above sum is equal to sin ± 2 π ( N + 1 2 ) x ² sin[ πx ] . for x 6 = 0. Hint: Use Euler’s formula and the identity 2 N X k =0 z k = ( z 2 N +1 - 1) / ( z - 1) . (2) Recall the delta function for the 3D integer lattice δ Z 3 can be approximated by N X h = - N N X k = - N N X l = - N e 2 πi ( hx + kk + lz ) . Show for x,y,z 6 = 0, the above sum is equal to f ( x ) f ( y ) f ( z ) where f ( x ) = sin ± 2 π ( N + 1 2 ) x ² sin[ πx ] . (3) (a) Show that Z 1 0 e 2 πikx e - 2 πijx dx = ³ 0 if j 6 = k 1 if j = k (b) Suppose f ( x ) = n X j = - n c j e 2 πijx . Use part a) to show that c k = Z 1 0 f ( x ) e - 2 πikx dx, for k = - n...n . (4) Consider the function f ( x ) = ´ 2 , if n - 1 4 < x < n + 1 4 , n an integer;

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Unformatted text preview: , otherwise. The Fourier coeﬃcient c j of the Fourier series ∑ ∞ j =-∞ c j e 2 πijx for f is given by c j = Z 1 / 2-1 / 2 f ( x ) e-2 πijx dx. Evaluate the integral to show that c = 1 and if j 6 = 0, c j = 0 if j is even and c j = 2(-1) k π (2 k + 1) 1 2 if j = 2 k + 1 is odd. Note: You can evaluate the integral by breaking it into real and imagi-nary parts, but you can also use the fact that the antiderivative of e-2 πijx is e-2 πijx / (-2 πij )....
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Homework3_11 - otherwise The Fourier coeﬃcient c j of the...

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