PHYS 408 HOMEWORK 6 SOLUTIONS

Jpg 83 function innity of delta functions 2 f x 0 hx

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Unformatted text preview: ￿ 8.2 ￿ dx￿ f (x￿ )h(x − x￿ ) = dx￿ f (x￿ )δ (x − x￿ − a) = f (x − a) (35) function, two delta functions ￿ dx￿ f (x￿ )h(x − x￿ ) = f (x − a) + f (x − (−a)) (36) Get the contribution we had in part (a) at the location a, as well a contribution from −a. Example plot for a = 4 and f (x) = exp(−x2 ) in figure 4. Figure 4: p8.jpg 8.3 function, infinity of delta functions 2 f (x) = 0 h(x) = +∞ ￿ n=−∞ 0<x<T (37) otherwise δ (x − (2n)T ) (38) So, for every integer n: 2 fn (x) = 0 0 < x − (2n)T < T otherwise 2 = 0 (2n)T < x− < (2n + 1)T otherwise Performing the sum over n we obtain the function presented in problem (2). 9 (39) 9 Convolution theorem Limits of integration are −∞ to +∞. F{f ∗ h} ￿￿ ￿ dx￿ f (x￿ )h(x − x￿ ) = dx￿ f (x￿ ) dx e−ikx h(x − x￿ ) ￿ ￿ ￿ ￿ ￿ ￿ = dx￿ f (x￿ ) d(x + x￿ ) e−ik(x+x ) h((x + x￿ ) − x￿ ) = dx￿ f (x￿ )e−ikx dx e−ikx h(x) = ￿ dx e−ikx ￿...
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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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