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jbay9009
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Homework Statement
(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
Homework Equations
The definition of the delta function: ∫(-inf to inf) δ(x-y)f(x)=f(y)
The Attempt at a Solution
(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 :)