The surface Green's function belonging to the non-spherical exterior boundary value problem of Helmholtz's equation in spherical coordinates is derived. This is performed in two ways, first by applying the Separation of Variables method, and, second, by using the Method of Lines as a special Finite-Difference technique. With this Green's function we are able to resolve some contradictions concerning conceptual aspects of the Separation of Variables method, the Finite-Difference methods, and the Boundary Integral Equation methods which have been developed for rigorously solving non-separable boundary value problems. The necessary mathematical background, the relation to Waterman's T matrix, and simplifications due to certain symmetry properties of the boundary surface will be discussed. In this paper we focus on the scalar problem. The extension to the vector case for electromagnetic wave scattering is in preparation and will be published later.
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