shaun_chou
Dec15-09, 06:30 PM
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?
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?