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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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