Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry

Published

March 23, 2012

Author(s)

Howard S. Cohl, Ernie G. Kalnins

Abstract

Due to the isotropy of d-dimensional hyperbolic space, there exists a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. The R-radius hyperboloid model of hyperbolic geometry with R > 0 represents a Riemannian manifold with negative-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 hyperbolic sine, finite summation expressions over hyperbolic functions, Gauss hypergeometric functions and in terms of the associated Legendre function of the second kind with order and degree given by d/2 − 1 with real argument greater than unity. We also demonstrate uniqueness for a fundamental solution of Laplace's equation on this manifold in terms of a vanishing decay at infinity. In rotationally invariant coordinate systems, we compute the azimuthal Fourier coefficients for a fundamental solution of Laplace's equation on the R-radius hyperboloid. For d≥2, we compute the Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace's equation on this negative-constant curvature Riemannian manifold. In three dimensions, an addition theorem for the azimuthal Fourier coefficients of a fundamental solution for Laplace's equation is obtained through comparison with its corresponding Gegenbauer expansion.

Citation

Journal of Physics A: Mathematical and Theoretical

Volume

45

Issue

14

Pub Type

Journals

Download Paper

https://doi.org/10.1088/1751-8113/45/14/145206

Keywords

Hyperbolic geometry, Fundamental solution, Laplace's equation, Fourier series, Gegenbauer series, Separation of variables

Citation

Cohl, H. and Kalnins, E. (2012), Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry, Journal of Physics A: Mathematical and Theoretical, [online], https://doi.org/10.1088/1751-8113/45/14/145206 (Accessed December 14, 2024)