Friday, February 3, 2012

1108.3679 (Howard S. Cohl)

Fundamental Solution of Laplace's Equation in Hyperspherical Geometry    [PDF]

Howard S. Cohl
Due to the isotropy of $d$-dimensional hyperspherical space, one expects
there to exist a spherically symmetric 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 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 associated Legendre function of the second kind on the cut
(Ferrers function of the second kind) with degree and order given by $d/2-1$
and $1-d/2$ respectively, with real argument between plus and minus one.
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