Image methods for constructing Green’s functions and eigenfunctions for domains with plane boundaries

Abstract

We consider the image problem for domains with plane boundaries. We list all three and two dimensional domains for which the image method yields solutions of the potential problem, and we describe the image arrays generated by these domains in familiar crystallographic terms. One obtains from the group–theoretic description of images two representations for the Dirichlet Green’s functions for ∇2. The first is obtained by summing the unrestricted Green’s function over the crystal image structures, and the second is obtained in terms of an eigenfunction expansion using solutions of ∇2ψ=λψ which vanish on the plane boundaries.