%% This LaTeX-file was created by <jl> Sat Mar 29 19:36:56 1997
%% LyX 0.10 (C) 1995 1996 by Matthias Ettrich and the LyX Team

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\begin{document}

Expansion in free space: 

\( d^{3}r=\rho d\rho d\phi dz \). So \( \delta (r^{i}-r'^{i})=\frac{1}{\rho }\delta (\rho -\rho ')\delta (\phi -\phi ')\delta (z-z') \)

This means \( G(r^{i},r^{i'})=\int _{-\infty }^{\infty }dke^{ikz}\sum _{m=-\infty }^{\infty }e^{im\phi }g_{m}(\rho ,r'^{i}) \) \( =\frac{1}{\pi }\int _{-\infty }^{\infty }dke^{ik(z-z')}\sum _{m=-\infty }^{\infty }e^{im(\phi -\phi ')}I_{m}(|k|\rho _{<})K_{m}(|k|\rho _{>}) \) \( =\frac{4}{\pi }\int _{0}^{\infty }dke^{ik(z-z')}\sum _{m=0}^{\infty }com(m(\phi -\phi '))(1-\frac{1}{2}\delta _{m0})I_{m}(k\rho _{<})K_{m}(k\rho _{>}) \)

\end{document}
