Homework3_11

Homework3_11 - , otherwise. The Fourier coecient c j of the...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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;
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , otherwise. The Fourier coecient c j of the Fourier series j =- c j e 2 ijx for f is given by c j = Z 1 / 2-1 / 2 f ( x ) e-2 ijx 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 (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 ijx is e-2 ijx / (-2 ij )....
View Full Document

Page1 / 2

Homework3_11 - , otherwise. The Fourier coecient c j of the...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online