View source for Toroidal harmonics

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Functions of the points on a torus that arise when solving the [[Laplace equation|Laplace equation]] by the method of separation of variables (cf. [[Separation of variables, method of|Separation of variables, method of]]) in [[Toroidal coordinates|toroidal coordinates]]  $  ( \sigma , \tau , \phi ) $. 
A [[Harmonic function|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,   A.A. Samarskii,   "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Morse,   H. Feshbach,   "Methods of theoretical physics" , '''1–2''' , McGraw-Hill  (1953)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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))</TD></TR></table>