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.
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Toroidal harmonics. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Toroidal_harmonics&oldid=17103