Inverse of the Jacobi elliptic function cd: Series representations (formula 09.37.06.0009)

InverseJacobiCD

$Mathematica Notation$

Elliptic Functions

InverseJacobiCD[z,m]

Series representations

Generalized power series

Expansions at generic point z==z₀

For the function itself

http://functions.wolfram.com/09.37.06.0009.01

Input Form

InverseJacobiCD[z, m] == InverseJacobiCD[Subscript[z, 0], m] - Sum[(1/k!) Sum[((Pochhammer[1 - k, 2 (k - j) - 2]/((k - j - 1)! (2 Subscript[z, 0])^(k - 2 j - 1))) Sum[Binomial[j, s] Pochhammer[1/2, s] Pochhammer[1/2, j - s] m^(j - s) (1 - Subscript[z, 0]^2)^(-(1/2) - s) (1 - m Subscript[z, 0]^2)^ (s - j - 1/2), {s, 0, j}]) (z - Subscript[z, 0])^k, {j, 0, k - 1}], {k, 1, Infinity}]

Standard Form

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MathML Form

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</mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "k"]], ")"]], RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]]]], "-", "2"]]], Pochhammer] </annotation> </semantics> <mtext> </mtext> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> s </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> j </mi> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> <mtr> <mtd> <mi> s </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> s </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "s"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - 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Rule Form

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

2007-05-02