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Conical Function

Functions which can be expressed in terms of Legendre functions of the first and second kinds. See Abramowitz and Stegun (1972, p. 337).

P_(-1/2+ip)(costheta)=1+(4p^2+1^2)/(2^2)sin^2(1/2theta)+((4p^2+1^2)(4p^2+3^2))/(2^24^2)sin^4(1/2theta)+...
(1)
=2/piint_0^theta(cosh(pt)dt)/(sqrt(2(cost-costheta)))
(2)
Q_(-1/2∓ip)(costheta)=+/-isinh(ppi)int_0^infty(cos(pt)dt)/(sqrt(2(cosht+costheta)))+int_0^infty(cosh(pt)dt)/(sqrt(2(cost-costheta))).
(3)

See also

Toroidal Function

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Conical Functions." §8.12 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 337, 1972.Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1464, 1980.

Referenced on Wolfram|Alpha

Conical Function

Cite this as:

Weisstein, Eric W. "Conical Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConicalFunction.html

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