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

SIGMA 11 (2015), 015, 23 pages      arXiv:1405.4847
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

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

Howard S. Cohl a and Rebekah M. Palmer b
a) Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA
b) Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA

Received May 20, 2014, in final form February 09, 2015; Published online February 14, 2015

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.

Key words: fundamental solution; hypersphere; Fourier expansion; Gegenbauer expansion.

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