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

Symmetry Integrability and Geometry-Methods and Applications

article

Howard S. Cohl¹ 

[1] Applied and Computational Mathematics Division, Information Technology Laboratory, National Institute of Standards and Technology;Department of Mathematics, University of Auckland

关键词: hyperspherical geometry;    opposite antipodal fundamental solution;    Laplace’s equation;    separation of variables;    Ferrers functions;   

DOI  :  10.3842/SIGMA.2011.108

来源: National Academy of Science of Ukraine

【 摘 要 】

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 $x\in(-1,1)$.

【 授权许可】

Unknown