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This article is cited in 3 scientific papers (total in 3 papers)
Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry
Howard S. Cohla, Rebekah M. Palmerb a Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA
b Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA
Abstract:
For a fundamental solution of Laplace's equation on the $R$-radius $d$-dimensional 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 three-dimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace's equation on the 3-sphere. Applications of our expansions are given, namely closed-form solutions to Poisson's equation with uniform density source distributions. The Newtonian potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given to the superintegrable Kepler–Coulomb and isotropic oscillator potentials.
Keywords:
fundamental solution; hypersphere; Fourier expansion; Gegenbauer expansion.
Received: May 20, 2014; in final form February 9, 2015; Published online February 14, 2015
Citation:
Howard S. Cohl, Rebekah M. Palmer, “Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry”, SIGMA, 11 (2015), 015, 23 pp.
Linking options:
https://www.mathnet.ru/eng/sigma996 https://www.mathnet.ru/eng/sigma/v11/p15
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