PHYS 408 HOMEWORK 6 SOLUTIONS

N1 2 6 kn n b 7 12 plot fourier decomposition

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online