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Internal and external harmonics in bi-cyclide coordinates



Hans Volkmer, Brandon Alexander, Howard Cohl


The Laplace equation in three-dimensional Euclidean space is R-separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lamé–Wangerin functions called internal and external bi-cyclide harmonics. An expansion for a fundamental solution of Laplace's equation in products of internal and external bi-cyclide harmonics is derived. In limiting cases this expansion reduces to known expansions in bi-spherical and prolate spheroidal coordinates.
Journal of Physics A: Mathematical and Theoretical


Laplace’s equation, fundamental solution, separable curvilinear coordinate system, bi-cyclide coordinates, Lamé-Wangerin functions.


Volkmer, H. , Alexander, B. and Cohl, H. (2023), Internal and external harmonics in bi-cyclide coordinates, Journal of Physics A: Mathematical and Theoretical, [online],, (Accessed April 17, 2024)
Created July 20, 2023, Updated March 27, 2024