### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 041, 31 pages      arXiv:2202.08918      https://doi.org/10.3842/SIGMA.2022.041

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

Lijuan Bi a, Howard S. Cohl b and Hans Volkmer c
a) Department of Mathematics,The Ohio State University at Newark,Newark, OH 43055, USA
b) Applied and ComputationalMathematics Division, National Institute of Standards and Technology, Mission Viejo, CA 92694, USA
c) Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201-0413, USA

Received November 20, 2021, in final form May 18, 2022; Published online June 03, 2022

Abstract
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.

Key words: Laplace's equation; fundamental solution; separable curvilinear coordinate system; flat-ring cyclide coordinates; special functions; orthogonal polynomials.

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