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Calculus and Beyond Homework Help
Integral containing a delta function and a step function
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[QUOTE="jbay9009, post: 4531102, member: 490507"] [h2]Homework Statement [/h2] (a) Show that that δ(a-b)=∫δ(x-a)δ(x-b)dx (b) Show that ∂/∂x θ(x) = δ(x) where θ(x) is the heaviside step function (0 for x<0, 1 for x>0) (c) Show that ∫(-inf to inf) δ(x) f(θ(x))dx=∫(0 to 1) f(y)dy [h2]Homework Equations[/h2] The definition of the delta function: ∫(-inf to inf) δ(x-y)f(x)=f(y) [h2]The Attempt at a Solution[/h2] (a) Just made a change of variables and compared to the definition of the δ-function (b) ∫∂/∂x θ(x)dx = θ(x)|limits = 1 if limits enclose 0, 0 if not = ∫δ(x)dx with same limits (c) I used the result from part (b) to get ∫(-inf to inf) ∂/∂x θ(x) f(θ(x))dx then integrated by parts to get θ(x) f(θ(x))|(-inf to inf) -∫(-inf to inf) θ(x) ∂/∂x f(θ(x)) = f(1) -∫(-inf to inf) θ(x) ∂/∂x f(θ(x)) Can anyone tell me if I'm going about this the right way? thanks in advance :) [/QUOTE]
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Integral containing a delta function and a step function
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