PHYS 408 HOMEWORK 6 SOLUTIONS

# N1 2 6 kn n b 7 12 plot fourier decomposition

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Unformatted text preview: ∞ 1￿ = E0 + sinc(π m/2) cos(ikn x) 2 n=1 2 (6) kn = πn b (7) 1.2 plot Fourier decomposition coeﬃcients Figure 1: p1.jpg 3 2 Another Fourier decomposition 2.1 Outright calculation Assume period of 2T and later adjust T as needed. 2 f (x) = 0 Fourier decompose f (x): +∞ ￿ f (x) = (2n)T < x < (2n + 1)T (8) (2n + 1)T < x < (2n + 2)T kn = cn eikn x n=−∞ 2π n πn = 2T T (9) Integrate over one period using the piecewise form of f (x): ￿ 2T −ikm x dx e f (x) = 0 ￿ T 0 dx e−ikm x · 2 = 2T · e−ikm T /2 sinc(km T /2) (10) Integrate over one period using the Fourier decomposition of f (x): ￿ 2T 0 dx e−ikm x f (x) = cm · 2T (11) Thus cm = e−ikm T /2 sinc(km T /2) (12) and f (x) = +∞ ￿￿ n=−∞ = 1+ ￿ e−ikn T /2 sinc(kn T /2) eikn x ￿￿ |n| odd = 1+2 ￿ n odd = 1+ (13) ￿ e−ikn T /2 sinc(kn T /2) eikn x (14) cos(kn x − kn T /2)sinc(kn T /2) 4 ￿ sin(kn x) π n (15) kn = n odd πn T (16) For T = 1, f (x) = 1 + 4 ￿ sin(nπ x) π n (17) n odd 2.2 Identify expansion in list (c) 2.3 Shortcuts f (x) takes the form of 1 plus an odd function, so we would expect the odd function part to consist...
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## This document was uploaded on 03/11/2014 for the course PHYS 408 at University of British Columbia.

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