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\begin{document}

The rectangular coordinate green function is \( G(r^{i},r'^{i})=\sum _{n,m}g_{nm}(z,r'^{i})sin(k_{xn}x)sin(k_{ym}y) \) where \( k_{xn}=\frac{n\pi }{a} \), and \( k_{ym}=\frac{m\pi }{b} \).

\( g_{nm} \) satisfies: \( (\frac{d^{2}}{dz^{2}}-k_{z,nm}^{2})g_{nm}(z,r'^{i})=-\frac{4}{ab}4\pi \delta (z-z')sin(k_{xn}x')sin(k_{ym}y') \)

so \( g_{nm}=h_{nm}(z,z')sin(k_{xn}x')sin(k_{ym}y') \) where: \( (\frac{d^{2}}{dz^{2}}-k_{z,nm}^{2})h_{nm}(z,z')=-\frac{1}{ab}16\pi \delta (z-z') \). Choose \( Asinh(k_{z,nm}z) \) for \( z<z' \) and \( Bsinh(k_{z,nm}(c-z)) \) for \( z>z' \).

so \( h_{nm}=Csinh(k_{z,nm}z_{min})sinh(k_{z,nm}(c-z_{max}) \)

leading to a total green's functions of: \( G(r^{i},r'^{i})=\frac{16\pi }{ab}\sum _{n,m}sin(k_{xn}x')sin(k_{xn}x)sin(k_{ym}y')sin(k_{ym}y)*\frac{1}{k_{z,nm}sinh(k_{z,nm}c)} \) \( *sinh(k_{z,nm}z_{min})sinh(k_{z,nm}(c-z_{max}) \)

The z coordinate has been given a special treatment. It'd be nice
to have a green function which is symmetric in all coordinates. This
can be done using a laplacian eigenfunction expansion. The rectangular
eigenfunctions are:

\( \psi _{lmn}(r^{i})=\frac{8}{abc}sin(k_{xl}x)sin(k_{ym}y)sin(k_{zn}z) \)

where \( k_{xl}=\frac{l\pi }{a} \)... and similar relations

This gives \( G(r^{i},r'^{i})=\frac{32}{abc\pi ^{2}}\sum _{l,n,m}\frac{1}{(\frac{l}{a})^{2}+(\frac{m}{b})^{2}+(\frac{n}{c})^{2}}\psi _{lmn}(r'^{i})\psi _{lmn}(r^{i}) \).

Comparing these green functions, you get: \( sinh(k_{z,lm}z_{min})sinh(k_{z,lm}(c-z))\frac{1}{k_{z,lm}sinh(k_{z,lm}c)}=\frac{2}{c}\sum _{n=1}^{\infty }\frac{sin(\frac{n\pi }{c}z)sin(\frac{n\pi }{c}z')}{k_{z,lm}^{2}+\frac{n^{2}\pi ^{2}}{c^{2}}} \)

which can be thought of as a fourier expansion or the series can just
be summed. 

\end{document}
