Inverse of the Jacobi elliptic function sd: Representations through equivalent functions (formula 09.47.27.0016)

InverseJacobiSD

$Mathematica Notation$

Elliptic Functions

InverseJacobiSD[z,m]

Representations through equivalent functions

With related functions

Involving elliptic integrals

http://functions.wolfram.com/09.47.27.0016.01

Input Form

InverseJacobiSD[z, m] == InverseJacobiSD[Subscript[z, 0], m] + ((Sqrt[1 + m z^2] JacobiCN[InverseJacobiSD[z, m], m])/ (Sqrt[1 - m] Sqrt[1 + (-1 + m) z^2])) (EllipticF[ArcSin[Sqrt[1 - m] z], m/(-1 + m)] - EllipticF[ArcSin[Sqrt[1 - m] Subscript[z, 0]], m/(-1 + m)]) /; !Exists[\[Tau], {Element[\[Tau], Reals], 0 < \[Tau] < 1}, Im[m (Subscript[z, 0] + \[Tau] (z - Subscript[z, 0]))^2 + 1] == 0 && m (Subscript[z, 0] + \[Tau] (z - Subscript[z, 0]))^2 + 1 < 0 && Im[1 - (1 - m) (Subscript[z, 0] + \[Tau] (z - Subscript[z, 0]))^2] == 0 && 1 - (1 - m) (Subscript[z, 0] + \[Tau] (z - Subscript[z, 0]))^2 < 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