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InverseJacobiSC




Mathematica Notation

Traditional Notation









Elliptic Functions > InverseJacobiSC[z,m] > Differentiation > Low-order differentiation > With respect to m





http://functions.wolfram.com/09.46.20.0004.01









  


  








Input Form





D[InverseJacobiSC[z, m], m] == (-(1/(2 (-1 + m) m (1 + z^2)))) ((1 + z^2) EllipticE[JacobiAmplitude[InverseJacobiSC[z, m], m], m] + (-1 + m) (1 + z^2) InverseJacobiSC[z, m] - m z JacobiND[InverseJacobiSC[z, m], m])








Standard Form





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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> <mo> &#8706; </mo> <mrow> <msup> <mi> sc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> m </mi> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> am </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10072; </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> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> sc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> nd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> m </ci> </bvar> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <ci> JacobiAmplitude </ci> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <ci> z </ci> <apply> <ci> JacobiND </ci> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <ci> m </ci> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["m_"]]], RowBox[List["InverseJacobiSC", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", "m"]], "]"]]]], "-", RowBox[List["m", " ", "z", " ", RowBox[List["JacobiND", "[", RowBox[List[RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", "m", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z", "2"]]], ")"]]]]]]]]]]]


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





2001-10-29