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

SIGMA 7 (2011), 108, 15 pages      arXiv:1108.3679      https://doi.org/10.3842/SIGMA.2011.108

Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

Howard S. Cohl a, b
a) Applied and Computational Mathematics Division, Information Technology Laboratory, National Institute of Standards and Technology, Mission Viejo, California, 92694 USA
b) Department of Mathematics, University of Auckland, 38 Princes Str., Auckland, New Zealand

Received August 18, 2011, in final form November 22, 2011; Published online November 29, 2011; Misprints are corrected January 29, 2012; Corrected December 28, 2018

Abstract
Due to the isotropy of $d$-dimensional hyperspherical space, one expects there to exist a spherically symmetric opposite antipodal fundamental solution for its corresponding Laplace-Beltrami operator. The $R$-radius hypersphere ${\mathbf S}_R^d$ with $R>0$, represents a Riemannian manifold with positive-constant sectional curvature. We obtain a spherically symmetric opposite antipodal fundamental solution of Laplace's equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the trigonometric sine, finite summation expressions over trigonometric functions, Gauss hypergeometric functions, and in terms of the Ferrers function of the second with degree and order given by $d/2-1$ and $1-d/2$ respectively, with real argument $x\in(-1,1)$.

Key words: hyperspherical geometry; opposite antipodal fundamental solution; Laplace's equation; separation of variables; Ferrers functions.

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