Elliptic Integral of the Third Kind
Let 0<k^2<1. The incomplete elliptic integral of the third kind is then defined as
Pi(n;phi,k) = [画像:int_0^phi(dtheta)/((1-nsin^2theta)sqrt(1-k^2sin^2theta))]
(1)
where n is a constant known as the elliptic characteristic and k is the elliptic modulus. It is implemented in the Wolfram Language as EllipticPi [n, phi, m].
EllipticPi
The complete elliptic integral of the third kind
| Pi(n|m)=Pi(n;1/2pi|m) |
(3)
|
is illustrated above.
See also
Complete Elliptic Integral of the Third Kind, Elliptic Integral of the First Kind, Elliptic Integral of the Second Kind, Elliptic Integral Singular Value, Elliptic ModulusRelated Wolfram sites
https://functions.wolfram.com/EllipticIntegrals/EllipticPi3/Explore with Wolfram|Alpha
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References
Abramowitz, M. and Stegun, I. A. (Eds.). "Elliptic Integrals" and "Elliptic Integrals of the Third Kind." Ch. 17 and §17.7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587-607, 1972.Tölke, F. "Normalintegrale dritter Gattung. Legendresche Pi-Funktion. Zurückführung des allgemeinen elliptischen Integrals auf Normalintegrale erster, zweiter, und dritter Gattung." Ch. 7 in Praktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 100-144, 1967.Referenced on Wolfram|Alpha
Elliptic Integral of the Third KindCite this as:
Weisstein, Eric W. "Elliptic Integral of the Third Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipticIntegraloftheThirdKind.html