# Toroidal harmonics

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.

#### References

[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) |

#### Comments

#### References

[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: http://www.encyclopediaofmath.org/index.php?title=Toroidal_harmonics&oldid=17103