Expansion for a Fundamental Solution of Laplace's Equation in Flat-Ring Cyclide Coordinates

TL;DR

This paper derives a new series expansion for the fundamental solution of Laplace's equation in flat-ring coordinates, using harmonic functions expressed via Lamé functions, and connects it to toroidal coordinates in a limit.

Contribution

It introduces a novel expansion of Laplace's fundamental solution in flat-ring coordinates using harmonic functions based on Lamé functions, extending coordinate system applications.

Findings

Derived a double series expansion for the fundamental solution

Expressed harmonic functions in terms of Lamé functions

Connected flat-ring and toroidal coordinate expansions

Abstract

We derive an expansion for the fundamental solution of Laplace's equation in flat-ring coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and exterior of "flat rings". These internal and external flat-ring harmonic functions are expressed in terms of simply-periodic Lam\'e functions. In a limiting case we obtain the expansion of the fundamental solution in toroidal coordinates.