Complete elliptic integral of the first kind: Series representations (formula 08.02.06.0034)

EllipticK

$Mathematica Notation$

Elliptic Integrals

EllipticK[z]

Series representations

Generalized power series

Expansions at z==infinity

For the function itself

http://functions.wolfram.com/08.02.06.0034.01

Input Form

EllipticK[z] == Subscript[F, Infinity][z] /; Subscript[F, n][z] == (1/(2 Sqrt[-z])) Sum[((Pochhammer[1/2, k]^2/k!^2) (2 PolyGamma[k + 1] - PolyGamma[1/2 + k] - PolyGamma[1/2 - k] + Log[-z]))/z^k, {k, 0, m}] == EllipticK[z] - (1/2) MeijerG[{{-(1/2) - m, -(1/2) - m}, {1/2, 1/2}}, {{0, -(1/2) - m, -(1/2) - m}, {0}}, -z] && Element[m, Integers] && m >= 0

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