Inverse of the Jacobi elliptic function cs: Representations through equivalent functions (formula 09.40.27.0005)

InverseJacobiDC

$Mathematica Notation$

Elliptic Functions

InverseJacobiDC[z,m]

Representations through equivalent functions

With related functions

Involving dn^-1

http://functions.wolfram.com/09.40.27.0005.01

Input Form

InverseJacobiDC[z, m] == (-I) InverseJacobiDN[z, 1 - m] + 2 I EllipticK[1 - m]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["InverseJacobiDC", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["InverseJacobiDN", "[", RowBox[List["z", ",", RowBox[List["1", "-", "m"]]]], "]"]]]], "+", RowBox[List["2", "\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> dc </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> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> dn </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> InverseJacobiDC </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> InverseJacobiDN </ci> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiDC", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["InverseJacobiDN", "[", RowBox[List["z", ",", RowBox[List["1", "-", "m"]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]]]]]]]]

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

2001-10-29