Expansion for a Fundamental Solution of Laplace's Equation in Flat-Ring Cyclide Coordinates

Lijuan Bi ; Howard S. Cohl ; Hans Volkmer

Symmetry, integrability and geometry: methods and applications, Tome 18 (2022)

Résumé

We derive an expansion for the fundamental solution of Laplace's equation in flat-ring coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and exterior of “flat rings”. These internal and external flat-ring harmonic functions are expressed in terms of simply-periodic Lamé functions. In a limiting case we obtain the expansion of the fundamental solution in toroidal coordinates.

Détail

Keywords: fundamental solution, separable curvilinear coordinate system, flat-ring cyclide coordinates, special functions
Mots-clés : Laplace's equation, orthogonal polynomials.

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