Incomplete elliptic integral of the first kind: Series representations (formula 08.05.06.0043)

EllipticF

$Mathematica Notation$

Elliptic Integrals

EllipticF[z,m]

Series representations

Generalized power series

Expansions at z==Pi/2+2Pi u/;u ∈ Z && m>1

http://functions.wolfram.com/08.05.06.0043.01

Input Form

EllipticF[z, m] \[Proportional] (-(1/Sqrt[m])) EllipticK[1/m] (I Sqrt[-(1/(z - Subscript[z, 0])^2)] (z - Subscript[z, 0]) + Sqrt[(z - Subscript[z, 0])^2]/(z - Subscript[z, 0])) + EllipticK[m] (2 Round[Re[Subscript[z, 0]]/Pi] + Sqrt[-(I/(z - Subscript[z, 0]))] Sqrt[I (z - Subscript[z, 0])] + I Sqrt[-(1/(z - Subscript[z, 0])^2)] (z - Subscript[z, 0]) + Sqrt[(z - Subscript[z, 0])^2]/(z - Subscript[z, 0])) + (z - Subscript[z, 0])/Sqrt[1 - m] - (m (z - Subscript[z, 0])^3)/ (6 (1 - m)^(3/2)) + \[Ellipsis] /; (z -> Subscript[z, 0]) && Subscript[z, 0] == Pi/2 + 2 Pi u && Element[u, Integers] && Element[m, Reals] && m > 1

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