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Elliptic Integral of the Second Kind


Let the elliptic modulus k satisfy 0<k^2<1. (This may also be written in terms of the parameter m=k^2 or modular angle alpha=sin^(-1)k.) The incomplete elliptic integral of the second kind is then defined as

 E(phi,k)=int_0^phisqrt(1-k^2sin^2theta)dtheta.
(1)

The elliptic integral of the second kind is implemented in the Wolfram Language as EllipticE[phi, m] (note the use of the parameter m=k^2 instead of the modulus k).

The complete elliptic integral of the second kind E(k) is defined by

 E(k)=E(1/2pi,k).
(2)

To place the elliptic integral of the second kind in a slightly different form, let

t=sintheta
(3)
dt=costhetadtheta=sqrt(1-t^2)dtheta,
(4)

so the elliptic integral can also be written as

E(phi,k)=int_0^(sinphi)sqrt(1-k^2t^2)(dt)/(sqrt(1-t^2))
(5)
=int_0^(sinphi)sqrt((1-k^2t^2)/(1-t^2))dt.
(6)

A generalization replacing sintheta with sinhtheta in (1) gives

 -iE(iphi,-k)=int_0^phisqrt(1-k^2sinh^2theta)dtheta.
(7)

The incomplete elliptic integral of the second kind of the form E(z,cscz) can be written in terms of complete elliptic integrals of the first K(k) and second kinds E(k) as

 E(z,cscz)=csczE(sinz)-coszcotzK(sinz)
(8)

for -pi/2<R[z]<pi/2.


See also

Complete Elliptic Integral of the Second Kind, Elliptic Integral of the First Kind, Elliptic Integral of the Third Kind, Elliptic Integral Singular Value

Related Wolfram sites

http://functions.wolfram.com/EllipticIntegrals/EllipticE2/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Elliptic Integrals." Ch. 17 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587-607, 1972.Spanier, J. and Oldham, K. B. "The Complete Elliptic Integrals K(p) and E(p)" and "The Incomplete Elliptic Integrals F(p;phi) and E(p;phi)." Chs. 61 and 62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 609-633, 1987.Tölke, F. "Parameterfunktionen." Ch. 3 in Praktische Funktionenlehre, zweiter Band: Theta-Funktionen und spezielle Weierstraßsche Funktionen. Berlin: Springer-Verlag, pp. 83-115, 1966.Tölke, F. "Umkehrfunktionen der Jacobischen elliptischen Funktionen und elliptische Normalintegrale erster Gattung. Elliptische Amplitudenfunktionen sowie Legendresche F- und E-Funktion. Elliptische Normalintegrale zweiter Gattung. Jacobische Zeta- und Heumansche Lambda-Funktionen," and "Normalintegrale dritter Gattung. Legendresche Pi-Funktion. Zurückführung des allgemeinen elliptischen Integrals auf Normalintegrale erster, zweiter, und dritter Gattung." Chs. 6-7 in Praktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 58-144, 1967.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

Referenced on Wolfram|Alpha

Elliptic Integral of the Second Kind

Cite this as:

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

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