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Homework Set 8 Math/ECE 430 due Friday, April 5, 2013 The page numbers below refer to the book by Gasquet and Witomski. R∞ 1. Show that hδ, φi = φ(0) implies that −∞ δ(x − y)φ(x)dx = φ(y) 2. Show that Rb a δ(x − y)dx = 1 provided a < y < b ; otherwise 3. Show that δ(ax) = Rb a δ(x − y)dx = 0. 1 δ(x). In particular, δ(−x) = δ(x). |a| 4. Show that δ(g(x)) = X δ(x − xj ) j |g 0 (xj )| , where xj are the roots of g(x). Here assume that g 0 is nonzero. 5. Show that H 0 = δ, where the prime denotes the weak or distributional derivative and H denotes the Heaviside function. 6. hδ 0 , φi =? 7. # 28.11 (solution of xT = 0 in D0 ) (a) Show that a function χ ∈ D satisfies χ(0) = 0 if and only if χ = xψ with ψ ∈ D. (Hint: think about Taylor expansions with remainder). (b) Let ζ ∈ D satisfy ζ(0) = 1. Show that for any φ ∈ D, there is some ψ ∈ D0 so that φ can be written as φ = φ(0)ζ + xψ. (Hint: You just need to show that χ = φ − φ(0)ζ is in D and is zero at the origin; then apply the above result. ) (c) Deduce from this that T ∈ D0 solves xT = 0 if and only if T = Kδ for some constant K. 8. # 28.12 (solution of xT = 1 in D0 .) Deduce from the above exercise that T solves xT = 1 if and only if T is of the form P.V.(1/x) + Kδ for some constant K. (Hint: We know that P.V.(1/x) is a solution; subtract it off to obtain the type of problem treated in problem 7.) 1