Toroidal functions are a class of functions also called ring functions that appear in systems having toroidal symmetry. Toroidal functions can be expressed in terms
of the associated Legendre functions
of the first and second kinds
(Abramowitz and Stegun 1972, p. 336):
for
as a "toroidal harmonic."
The toroidal functions are solutions to the differential equation for Laplace's
equation in toroidal coordinates .
See also Associated Legendre Polynomial ,
Conical Function ,
Laplace's
Equation--Toroidal Coordinates
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References Abramowitz, M. and Stegun, I. A. (Eds.). "Toroidal Functions (or Ring Functions)." §8.11 in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Byerly, W. E. An
Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal
Harmonics, with Applications to Problems in Mathematical Physics. Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic
Dictionary of Mathematics. Referenced
on Wolfram|Alpha Toroidal Function
Cite this as:
Weisstein, Eric W. "Toroidal Function."
From MathWorld https://mathworld.wolfram.com/ToroidalFunction.html
Subject classifications
![P_(nu-1/2)^mu(cosheta)=[Gamma(1-mu)]^(-1)2^(2mu)(1-e^(-2eta))^(-mu)e^(-(nu+1/2)eta)_2F_1(1/2-mu,1/2+nu-mu;1-2mu;1-e^(-2eta))
P_(n-1/2)^m(cosheta)=(Gamma(n+m+1/2)(sinheta)^m)/(Gamma(n-m+1/2)2^msqrt(pi)Gamma(m+1/2))int_0^pi(sin^(2m)phidphi)/((cosheta+cosphisinheta)^(n+m+1/2))
Q_(nu-1/2)^mu(cosheta)=[Gamma(1+nu)]^(-1)sqrt(pi)e^(imupi)Gamma(1/2+nu+mu)(1-e^(-2eta))^mue^(-(nu+1/2)eta)_2F_1(1/2-mu,1/2+nu+mu;1+mu;1-e^(-2eta))
Q_(n-1/2)^m(cosheta)=((-1)^mGamma(n+1/2))/(Gamma(n-m+1/2))int_0^infty(cosh(mt)dt)/((cosheta+coshtsinheta)^(n+1/2))](/images/equations/ToroidalFunction/NumberedEquation1.svg)
.
Byerly (1959) identifies
of Laplace's
equation in toroidal coordinates.
