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 .
Byerly (1959) identifies

as a "toroidal harmonic."

The toroidal functions are solutions to the differential equation for of 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.
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. Referenced
on Wolfram|Alpha Toroidal Function
Cite this as:
Weisstein, Eric W. "Toroidal Function."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/ToroidalFunction.html

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