Inverse of the Jacobi elliptic function cd: Differentiation (formula 09.37.20.0012)

InverseJacobiCD

$Mathematica Notation$

Elliptic Functions

InverseJacobiCD[z,m]

Differentiation

Low-order differentiation

With respect to m

http://functions.wolfram.com/09.37.20.0012.01

Input Form

D[InverseJacobiCD[z, m], {m, 3}] == ((-1 + m z^2) ((-8 - 23 (-1 + m) m) EllipticE[JacobiAmplitude[ InverseJacobiCD[z, m], m], m] - (-1 + m) (-7 + 11 m) EllipticF[JacobiAmplitude[InverseJacobiCD[z, m], m], m]) - 15 (-1 + m)^3 (-1 + m z^2) InverseJacobiCD[z, m] - m z^2 ((-1 + m)/(-1 + m z^2))^(3/2) (4 + m (-8 + (-7 + 11 m) z^2)) JacobiSC[InverseJacobiCD[z, m], m])/(4 (-1 + m)^3 m^3 (-2 + 2 m z^2))

Standard Form

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MathML Form

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</mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> cd </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 23 </mn> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> cd </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 11 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> cd </mi> <mrow> <mo> - 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Rule Form

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

2007-05-02