Incomplete elliptic integral of the second kind: Series representations (formula 08.04.06.0086)

EllipticE

$Mathematica Notation$

Elliptic Integrals

EllipticE[z,m]

Series representations

Other series representations

Expansions E(sin^-1(z)|m) at z==infinity

http://functions.wolfram.com/08.04.06.0086.01

Input Form

EllipticE[ArcSin[z], m] \[Proportional] (m z Sqrt[-z^2])/Sqrt[(-m) z^2] + ((Sqrt[-z^2] Sqrt[-m])/(2 z)) ((((I Sqrt[-z^2])/Sqrt[z^2]) (Sqrt[1/m] - 1/Sqrt[m]) - Sqrt[1/m] - 1/Sqrt[m]) EllipticE[m] + 2 EllipticE[1/m] + 2 ((1 - m)/m) EllipticK[1/m] + 2 I (1 - Sqrt[m/(m - 1)] Sqrt[(m - 1)/m]) (-EllipticE[1 - 1/m] + (1/m) EllipticK[1 - 1/m])) + (((1 - m) Sqrt[-z^2])/(2 z Sqrt[(-m) z^2])) (1 + O[1/z^2])

Standard Form

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MathML Form

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Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02

EllipticE[z]