View source for Toroidal harmonics

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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932801.png" />. A [[Harmonic function|harmonic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932802.png" />, which is a solution of the Laplace equation, can be written as a series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932803.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932804.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932805.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932807.png" /> are the associated [[Legendre functions|Legendre functions]] with half-integer index. By setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932808.png" /> one obtains a toroidal harmonic or a surface toroidal harmonic, this in contrast with the expression (*) which, as a function of the three variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932809.png" />, 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t09328010.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t09328011.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t09328012.png" /> 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>