Fourier and Gegenbauer expansions for a fundamental solution of Laplace's equation in hyperspherical geometry

Published

February 14, 2015

Author(s)

Howard S. Cohl, Rebekah M. Palmer

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, and give an integral representation for the higher 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. In an appendix we derive integral representations for associated Legendre and Ferrers functions of the first and second kind with degree and order equal to within a sign.

Citation

Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

Volume

015

Pub Type

Journals

Download Paper

https://doi.org/10.3842/SIGMA.2015.015

Keywords

Fundamental solution, Hypersphere, Fourier expansion, Gegenbauer expansion

Citation

Cohl, H. and Palmer, R. (2015), Fourier and Gegenbauer expansions for a fundamental solution of Laplace's equation in hyperspherical geometry, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), [online], https://doi.org/10.3842/SIGMA.2015.015 (Accessed October 11, 2025)