Proving Fourier Bessel Series Expansion with Limiting Procedures

[cabin5]

Homework Statement

By appropriate limiting procedures prove the following expansion
$$\frac{1}{\left(\rho^2+z^2\right)^{1/2}}=\int^{\infty}_{0} e^{-k\left|z\right|}J_{0}(k\rho)dk$$

Homework Equations

The Attempt at a Solution

I tried to implicate the fourier-bessel series but it turned out that there is no k dependence on coefficients of the series.

What can I do to proceed this question?

 

[Thaakisfox]

I think you should write out the integral representation of this 0-order bessel function, and then switch the integrals. With this you get a 2D Fourier transform of the given function.
After all, the Hankel-transform with zero order bessel functions, is the 2D Fourier transform of some cylindricaly symmetric two variable function.