How Does the Delta Function and Its Derivative Interact with Shifted Functions?

[gimak]

Homework Statement

Prove the following
'()( − ) = −′()
∫_-∞^∞δ'(x)*f(x-a) = -f'(a)

Homework Equations

∫_-∞^∞δ'(x-a)*f(x) = f(a)

The Attempt at a Solution

[/B]
∫_-∞ ^∞δ'(x)*f(x-a) = ∫δ(x)*f(x-a)dx-∫f'(x-a)*δ(x)dx = f(-a) - f'(-a)
Went from 1st to second by integration by parts
Used integral definition of delta function to go to 3rd part

 

[gimak]

The Attempt at a Solution

1) ∫-∞ ∞δ'(x)*f(x-a)
2) = ∫δ(x)*f(x-a)dx-∫f'(x-a)*δ(x)dx
3) = f(-a) - f'(-a)
Went from 1st to second by integration by parts
Used integral definition of delta function to go to 3rd part

Made above easier to read