Internal and external harmonics in bi-cyclide coordinates

TL;DR

This paper explores solutions to the Laplace equation using bi-cyclide coordinates, deriving an expansion for the fundamental solution in terms of internal and external harmonics, connecting to known coordinate systems.

Contribution

It introduces a new expansion for the fundamental solution of Laplace's equation using bi-cyclide harmonics, bridging to bi-spherical and prolate spheroidal cases.

Findings

Derived an expansion for the fundamental solution in bi-cyclide coordinates.

Connected the expansion to known solutions in bi-spherical and prolate spheroidal coordinates.

Enhanced understanding of harmonic functions in complex coordinate systems.

Abstract

The Laplace equation in three dimensional Euclidean space is $R$-separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lam\'e-Wangerin functions called internal and external bi-cyclide harmonics. An expansion for the 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 expansion in bi-spherical and prolate spheroidal coordinates.