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Date: 19-5-2019
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Date: 12-8-2018
1572
Date: 15-5-2018
1897
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Let . The incomplete elliptic integral of the third kind is then defined as
(1) |
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(2) |
where is a constant known as the elliptic characteristic and is the elliptic modulus. It is implemented in the Wolfram Language as EllipticPi[n, phi, m].
The complete elliptic integral of the third kind
(3) |
is illustrated above.
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 -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.
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