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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 :)