satisfies: 
where 
, and 
.
satisfies:
so 
where: 
. Choose 
for 
and 
for 
.
so 
leading to a total green's functions of: 
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:
where 
... and similar relations
This gives 
.
Comparing these green functions, you get: 
which can be thought of as a fourier expansion or the series can just be summed.