Internal and external harmonics in bi-cyclide coordinates

Abstract

The Laplace equation in three-dimensional Euclidean space is -separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lamé–Wangerin functions called internal and external bi-cyclide harmonics. An expansion for a fundamental solution of Laplace’s equation in products of internal and external bi-cyclide harmonics is derived. In limiting cases this expansion reduces to known expansions in bi-spherical and prolate spheroidal coordinates.