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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)=int_0^phi(dtheta)/((1-nsin^2theta)sqrt(1-k^2sin^2theta))
(1)
=int_0^(sinphi)(dt)/((1-nt^2)sqrt((1-t^2)(1-k^2t^2))),
(2)

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 Modulus

Related Wolfram sites

http://functions.wolfram.com/EllipticIntegrals/EllipticPi3/

Explore with Wolfram|Alpha

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 Kind

Cite this as:

Weisstein, Eric W. "Elliptic Integral of the Third Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipticIntegraloftheThirdKind.html

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