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 $ ( \sigma , \tau , \phi ) $. A harmonic function $ h = h ( \sigma , \tau , \phi ) $, which is a solution of the Laplace equation, can be written as a series

$$ \tag{* } h = \sqrt {\cosh \tau - \cos \sigma } \times $$

$$ \times \sum _ {j, k = 0 } ^ \infty [ A _ {jk} P _ {j - 1/2 } ^ {(} k) ( \cosh \tau ) + B _ {jk} Q _ {j - 1/2 } ^ {(} k) ( \cosh \tau )] \times $$

$$ \times ( a _ {k} \cos k \sigma + b _ {k} \sin k \sigma ) ( c _ {j} \cos j \phi + d _ {j} \sin j \phi ), $$

where the $ P _ {j - 1/2 } ^ {(} k) $, $ Q _ {j - 1/2 } ^ {(} k) $ are the associated Legendre functions with half-integer index. By setting $ \tau = \tau _ {0} $ one obtains a toroidal harmonic or a surface toroidal harmonic, this in contrast with the expression (*) which, as a function of the three variables $ ( \sigma , \tau , \phi ) $, 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

$$ { \frac{1}{\sqrt {\cosh \tau - \cos \sigma } } } = $$

$$ = \ { \frac{\sqrt 2 } \pi } \left ( Q _ {- 1/2 } ( \cosh \tau ) + 2 \sum _ {k = 1 } ^ \infty Q _ {k - 1/2 } ( \cosh \tau ) \cos k \sigma \right ) , $$

where $ Q _ {k - 1/2 } $ 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)

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))