Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry
Abstract
For a fundamental solution of Laplace's equation on the Rradius ddimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geodesic polar coordinates. From this expansion in threedimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace's equation on the 3sphere. Applications of our expansions are given, namely closedform solutions to Poisson's equation with uniform density source distributions. The Newtonian potential is obtained for the 2disc on the 2sphere and 3ball and circular curve segment on the 3sphere. Applications are also given to the superintegrable KeplerCoulomb and isotropic oscillator potentials.
 Publication:

SIGMA
 Pub Date:
 February 2015
 DOI:
 10.3842/SIGMA.2015.015
 arXiv:
 arXiv:1405.4847
 Bibcode:
 2015SIGMA..11..015C
 Keywords:

 fundamental solution;
 hypersphere;
 Fourier expansion;
 Gegenbauer expansion;
 Mathematics  Classical Analysis and ODEs;
 Mathematical Physics;
 Mathematics  Differential Geometry;
 31C12;
 32Q10;
 33C05;
 33C45;
 33C55;
 35J05;
 35A08;
 42A16
 EPrint:
 SIGMA 11 (2015), 015, 23 pages