The green's function for the poisson equation is defined by . Using greens theorem, you get: .

This means that if S goes to infinity, the solution will return to the equation for the scalar potential without boundary conditions.

For the dirichlet case, want to get rid of 2nd integral. So want on the surface. Then . If S is a perfect conductor, then last term becomes . .

For the neumann problem, choose, on the boundary. Then . so .

source psfile jl@crush.caltech.edu index rectangular_coordinate_green_function