Delta-function representations, Bessel function

Question

I know there is a $\delta$-function representation in terms of the Bessel function as

$$\delta(x-a) = x\int_0^{\infty} t J_{\nu}(xt)J_{\nu}(at)\; dt$$

Is there something similar for

$$\int_0^{\infty} t J_{\mu}(xt)J_{\nu}(at)\; dt$$

Maybe there is something I can do to at least get a reasonable series representation... All I can find are the integrals where the power of $t$ is related in a certain way to the orders $\mu$ and $\nu$, but it is not the case in my case. Thanks!

Answer 1

The integral you seek is null for $\mu\ne\nu$. See the book by Smythe, "Static and dynamic electricity", sections 5.296-5.298.