Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

Abstract

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 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.

Keywords

• hyperspherical geometry
• fundamental solution
• Laplace's equation
• separation of variables
• Ferrers functions