Integration of the product of sine and the first Bessel function

[Ardat]

Homework Statement

I'm supposed to prove that:

\int_0^∞[STRIKE][/STRIKE]sin(ka)J₀(kp)dk = (a² - p²)^1/2 if p < a
and = 0 if p > a

J₀ being the first Bessel function.

Homework Equations

The Attempt at a Solution

I've tried to inverse the order of integration and then make the integral form of the delta Dirac function appear, but I'm not sure how to do it, and so far my attemps have failed.

I also tried to put both sine and the bessel function as their series form, then transfom the infinite series into a limit of a finite series so I can interchange the sum and the integral, but it doesn't really leads me anywhere.

If anyone could give me some advise on how to resolve this, I would be grateful.

 

[Ardat]

Nobody can help me?