Expansions for a fundamental solution of Laplace's equation on ${\mathbb{R}}^3$ in 5-cyclidic harmonics

Abstract

We derive eigenfunction expansions for a fundamental solution of Laplace's equation in three-dimensional Euclidean space in 5-cyclidic coordinates. There are three such expansions in terms of internal and external 5-cyclidic harmonics of first, second and third kind. The internal and external 5-cyclidic harmonics are expressed by solutions of a Fuchsian differential equation with five regular singular points.

References in corpus (3)

• Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry
• Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems
• Eigenfunction expansions for a fundamental solution of Laplace's equation on ${\mathbb{R}}^3$ in parabolic and elliptic cylinder coordinates