- #1
Pengwuino
Gold Member
- 5,123
- 20
Homework Statement
By appropriate limiting procedures prove the following expansion:
[tex]J_0 (k\sqrt {\rho ^2 + \rho '^2 - 2\rho \rho '\cos (\phi )} ) = \sum\limits_{m = - \infty }^\infty {e^{im\phi } J_m (k\rho )J_m (k\rho ')} [/tex]
Homework Equations
[tex]\frac{1}{{|\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
\over x} - \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
\over x} '|}} = \frac{1}{{\sqrt {\rho ^2 + \rho '^2 - 2\rho \rho '\cos (\phi - \phi ') + (z - z')^2 } }}[/tex]
[tex]\[
= \int\limits_0^\infty {\sum\limits_{m = - \infty }^\infty {J_m (k\rho )J_m (k\rho ')} e^{im(\phi - \phi ')} e^{ - k(z_ > - z_ < )} dk}
\]
[/tex]
The Attempt at a Solution
I can't even really figure out what is going on at this part. Apparently something has rid the expansion of the integral, and the only thing that could do that is a delta function in k. I don't under what situation we are going to just have a 0th order bessel function either. This question is quite the stumper for me... any help would be appreciated :)