Toroidal Function

A class of functions also called Ring Functions which appear in systems having toroidal symmetry. Toroidal functions can be expressed in terms of the Legendre Functions and Second Kinds (Abramowitz and Stegun 1972, p. 336):

$P_{\nu-1/2}^\mu(\cosh\eta)=[\Gamma(1-\mu)]^{-1}2^{2\mu}(1-e^{-2\eta})^{-\mu}e^{-(\nu+1/2)\eta}$
$\times\, {}_2F_1({\textstyle{1\over 2}}-\mu, {\textstyle{1\over 2}}+\nu-\mu; 1-2\mu; 1-e^{-2\eta})$
$P_{n-1/2}^m(\cosh\eta)={\Gamma(n+m+{\textstyle{1\over 2}})(\sinh\eta)^m\over\Ga... ...\int_0^{\pi} {sin^{2m}\phi\,d\phi\over (\cosh\eta+\cos\phi\sinh\eta)^{n+m+1/2}}$
$Q_{\nu-1/2}^\mu(\cosh\eta)=[\Gamma(1+\nu)]^{-1}\sqrt{\pi}\,e^{i\mu\pi}\Gamma({\textstyle{1\over 2}}+\nu+\mu)$
$\times\, (1-e^{-2\eta})^\mu e^{-(\nu+1/2)\eta}{}_2F_1({\textstyle{1\over 2}}-\mu, {\textstyle{1\over 2}}+\nu+\mu; 1+\mu; 1-e^{-2\eta})$
$Q_{n-1/2}^m(\cosh\eta)={(-1)^m\Gamma(n+{\textstyle{1\over 2}})\over\Gamma(n-m+{... ...er 2}})}\int_0^\infty {\cosh(mt)\,dt\over (\cosh\eta+\cosh t\sinh\eta)^{n+1/2}}$

for $n>m$. Byerly (1959) identifies

$${1\over i^{n/2}}P_m^n(\coth x)=\mathop{\rm csch}\nolimits ^n x {d^nP_m(\coth x)\over d(\coth x)^n}$$

as a Toroidal Harmonic.

See also Conical Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Toroidal Functions (or Ring Functions).'' §8.11 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 336, 1972.

Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 266, 1959.

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1468, 1980.