Inverse of the Jacobi elliptic function cs: Differentiation (formula 09.40.20.0012)

InverseJacobiDC

$Mathematica Notation$

Elliptic Functions

InverseJacobiDC[z,m]

Differentiation

Low-order differentiation

With respect to m

http://functions.wolfram.com/09.40.20.0012.01

Input Form

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

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["m", ",", "3"]], "}"]]], RowBox[List["InverseJacobiDC", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "8"]], "-", RowBox[List["23", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", "m"]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiDC", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["11", " ", "m"]]]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiDC", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]]]], ")"]]]], "-", RowBox[List["15", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"], " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["InverseJacobiDC", "[", RowBox[List["z", ",", "m"]], "]"]]]], "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]], RowBox[List["m", "-", SuperscriptBox["z", "2"]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["11", " ", SuperscriptBox["m", "2"]]], "+", RowBox[List["4", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["m", " ", RowBox[List["(", RowBox[List["7", "+", RowBox[List["8", " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["JacobiSC", "[", RowBox[List[RowBox[List["InverseJacobiDC", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["8", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"], " ", SuperscriptBox["m", "3"], " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mn> 3 </mn> </msup> <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> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo>  </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 11 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sc </mi> <mo> ⁡ </mo> <mo> ( </mo> <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> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </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> 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> <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> 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> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 15 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <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> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> m </ci> <degree> <cn type='integer'> 3 </cn> </degree> </bvar> <apply> <ci> InverseJacobiDC </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 11 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 7 </cn> </apply> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> JacobiSC </ci> <apply> <ci> InverseJacobiDC </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -23 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <ci> m </ci> </apply> <cn type='integer'> -8 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <ci> JacobiAmplitude </ci> <apply> <ci> InverseJacobiDC </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 11 </cn> <ci> m </ci> </apply> <cn type='integer'> -7 </cn> </apply> <apply> <ci> EllipticF </ci> <apply> <ci> JacobiAmplitude </ci> <apply> <ci> InverseJacobiDC </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> InverseJacobiDC </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["m_", ",", "3"]], "}"]]]]], RowBox[List["InverseJacobiDC", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "8"]], "-", RowBox[List["23", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", "m"]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiDC", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["11", " ", "m"]]]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiDC", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]]]], ")"]]]], "-", RowBox[List["15", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"], " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["InverseJacobiDC", "[", RowBox[List["z", ",", "m"]], "]"]]]], "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", SuperscriptBox["z", "2"]]], RowBox[List["m", "-", SuperscriptBox["z", "2"]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["11", " ", SuperscriptBox["m", "2"]]], "+", RowBox[List["4", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["m", " ", RowBox[List["(", RowBox[List["7", "+", RowBox[List["8", " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["JacobiSC", "[", RowBox[List[RowBox[List["InverseJacobiDC", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]]]], RowBox[List["8", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"], " ", SuperscriptBox["m", "3"], " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["m", "-", SuperscriptBox["z", "2"]]], ")"]]]]]]]]]

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

2007-05-02