# Expansion of free space Green function in Bessel function

1. Dec 15, 2009

### shaun_chou

1. The problem statement, all variables and given/known data
In Jackson 3.16 we have to prove the expansion $$\frac{1}{\left{|}\vec{x}-\vec{x'}\right{|}}=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}dke^{im(\phi-\phi')}J_m(k\rho)J_m(k\rho')e^{-k(z_{>}-z_{<})}$$

2. Relevant equations

3. The attempt at a solution
I tried to use the techniques in the text book but I only got $$G=\frac{1}{2}\sum e^{im(\phi-phi')}\int_0^{\infty}dke^{k(z-z'}N_m(k\rho_>)J_m(k\rho_<)$$I just can't get the relationship of $$z_>$$ and $$z_<$$ in the expansion. Can anybody help me?