Elliptic Integral of the Third Kind
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"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
الجزء والصفحة:
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25-4-2019
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Elliptic Integral of the Third Kind
Let 
. The incomplete elliptic integral of the third kind is then defined as
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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