# Homework Help: Fourier Bessel series

1. Jan 6, 2009

### cabin5

1. The problem statement, all variables and given/known data
By appropriate limiting procedures prove the following expansion
$$\frac{1}{\left(\rho^2+z^2\right)^{1/2}}=\int^{\infty}_{0} e^{-k\left|z\right|}J_{0}(k\rho)dk$$

2. Relevant equations

3. 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?

2. Jan 7, 2009

### 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.