Toroidal harmonics

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Functions of the points on a torus that arise when solving the Laplace equation by the method of separation of variables (cf. Separation of variables, method of) in toroidal coordinates . A harmonic function , which is a solution of the Laplace equation, can be written as a series


where the , are the associated Legendre functions with half-integer index. By setting one obtains a toroidal harmonic or a surface toroidal harmonic, this in contrast with the expression (*) which, as a function of the three variables , is sometimes called a spatial toroidal harmonic.

The series (*) is used in the solution of boundary value problems in toroidal coordinates, taking into account the expansion

where is the Legendre function of the second kind.


[1] A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[2] P.M. Morse, H. Feshbach, "Methods of theoretical physics" , 1–2 , McGraw-Hill (1953)



[a1] H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1. The gamma function. The hypergeometric functions. Legendre functions , McGraw-Hill (1953) (Formula 3.10 (3))
How to Cite This Entry:
Toroidal harmonics. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098