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Date: 3-6-2019
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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.
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.
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