Complete elliptic integral of the third kind: Representations through more general functions (formula 08.03.26.0002)

EllipticPi

$Mathematica Notation$

Elliptic Integrals

EllipticPi[n,m]

Representations through more general functions

Through Meijer G

Classical cases

http://functions.wolfram.com/08.03.26.0002.01

Input Form

EllipticPi[1 - 1/Sqrt[z], 1 - 1/z] == ((Sqrt[z] + 1)/2) EllipticK[1 - 1/z] - (1/(4 Pi)) (EllipticK[1 - 1/z] MeijerG[{{1/2, 3/2}, {}}, {{0, 1}, {}}, z] + EllipticE[1 - 1/z] MeijerG[{{3/2, 3/2}, {}}, {{1, 1}, {}}, z]) /; !IntervalMemberQ[Interval[{-Infinity, -1}], z]

Standard Form

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MathML Form

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</mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29

EllipticPi[n,z,m]