# Green's function expansion in a set of eigenfunction

by kristobal hunta
Tags: eigenfunction, expansion, function, green
 P: 7 Hi! I encountered the problem that I need to decompose the Green function into a set of eigenfunction. Particularly, I have the free space Green function $$G(\vec r; \vec r') = \frac {e^{i k | \vec r - \vec r'|} } {4 \pi | \vec r - \vec r'|}$$ and I need to express it into series of cylindrical mode eigenfunctions $$\Psi ( \vec r; k) = H_m ( q r) sin( h z) e^{i m \phi}$$ $$k^2 = q^2 + h^2, h = \frac { \pi } {2 L}$$ here H - Hankel's function of the first kind. Eigenfunction forms a complete set, with discrete spectrum of eigenvalues q and h. I know that we can decompose the Green function into set of eigenfunctions, but I have the Green function for spherical representation, and eigenfunctions are from waveguide formed by two infinite plates parallel to each other. I couldn't find anything relevant about expanding the Green function into arbitrary set of eigenfunctions. Would appreciate any opinion or advice on the matter :)