Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

TL;DR

This paper derives explicit Fourier and Gegenbauer expansions for fundamental solutions of Laplace's equation on hyperspheres, enabling closed-form solutions for related potential problems in higher-dimensional spherical geometries.

Contribution

It provides new closed-form Fourier and Gegenbauer expansions for Laplace's fundamental solutions on hyperspheres, including an addition theorem in three dimensions.

Findings

Closed-form Fourier coefficients in 2D and 3D

Gegenbauer polynomial expansion for hyperspherical Laplace solutions

Applications to Poisson, Kepler-Coulomb, and oscillator potentials

Abstract

For a fundamental solution of Laplace's equation on the $R$-radius $d$-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geodesic polar coordinates. From this expansion in three-dimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace's equation on the 3-sphere. Applications of our expansions are given, namely closed-form solutions to Poisson's equation with uniform density source distributions. The Newtonian potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given to the superintegrable Kepler-Coulomb and isotropic oscillator potentials.