# PHYS 598 hw3 - 1 Test functions and distributions Read the...

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Physics 498/MMA Handout 3 Fall 2007 Mathematical Methods in Physics I m-stone5 Prof. M. Stone 2117 ESB University of Illinois 1) Test functions and distributions : Read the sections on distributions in chapter two of the lecture notes, then do the following problems: a) Let f ( x ) be a smooth function. i) Show that f ( x ) δ ( x ) = f (0) δ ( x ). Deduce that d dx [ f ( x ) δ ( x )] = f (0) δ 0 ( x ) . ii) We might also have used the product rule to conclude that d dx [ f ( x ) δ ( x )] = f 0 ( x ) δ ( x ) + f ( x ) δ 0 ( x ) . By integrating both against a test function, show this expression for the derivative of f ( x ) δ ( x ) is equivalent to that in part i). b) Let ϕ ( x ) be a test function. Using the definition of the principal part integrals , show that ∂t P Z -∞ ϕ ( x ) ( x - t ) dx = P Z -∞ ϕ ( x ) - ϕ ( t ) ( x - t ) 2 dx in two different ways: i) Fix the value of the cutoff . Differentiate the resulting -regulated integral, taking care to include the terms arising from the t dependence of the limits at x = t ± .

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• Fall '07
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