TOPICS
Elliptic Integral of the Third Kind
Let
. The incomplete elliptic integral of the third kind is then defined as
 |  |  | (1) |
 |  |  | (2) |
where
is a constant known as theelliptic characteristic and
is theelliptic modulus. It is implemented in theWolfram Language asEllipticPi[n,phi,m].
Thecomplete elliptic integralof the third kind
 | (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
http://functions.wolfram.com/EllipticIntegrals/EllipticPi3/Explore with Wolfram|Alpha
More things to try:
References
Abramowitz, M. and Stegun, I. A. (Eds.). "Elliptic Integrals" and "Elliptic Integrals of the Third Kind." Ch. 17 and §17.7 inHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587-607, 1972.Tölke, F. "Normalintegrale dritter Gattung. Legendresche
-Funktion. Zurückführung des allgemeinen elliptischen Integrals auf Normalintegrale erster, zweiter, und dritter Gattung." Ch. 7 inPraktische 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 theThird Kind." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/EllipticIntegraloftheThirdKind.html