D[InverseJacobiDC[z, m], m] == (1/(2 m (1 - m))) ((z Sqrt[1 - z^2])/Sqrt[m - z^2] + Sqrt[m] (EllipticE[1/m] - EllipticE[ArcSin[z], 1/m])) /; Element[z, Reals] && z^2 > 1 && z^2 - m > 0 && m < 1

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "m"], RowBox[List["InverseJacobiDC", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", "m", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]]]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List["m", "-", SuperscriptBox["z", "2"]]]]], "+", RowBox[List[SqrtBox["m"], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", FractionBox["1", "m"], "]"]], "-", " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcSin", "[", "z", "]"]], ",", FractionBox["1", "m"]]], "]"]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "\[And]", RowBox[List[SuperscriptBox["z", "2"], ">", "1"]], "\[And]", RowBox[List[RowBox[List[SuperscriptBox["z", "2"], "-", "m"]], ">", "0"]], "\[And]", RowBox[List["m", "<", "1"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <mo> &#8706; </mo> <mrow> <msup> <mi> dc </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> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <msqrt> <mrow> <mi> m </mi> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> + </mo> <mrow> <msqrt> <mi> m </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10072; </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mi> m </mi> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> m </ci> </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'> 2 </cn> <ci> m </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <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='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <ci> EllipticE </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticE </ci> <apply> <arcsin /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> z </ci> <reals /> </apply> <apply> <gt /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["m_"]]], RowBox[List["InverseJacobiDC", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[FractionBox[RowBox[List["z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List["m", "-", SuperscriptBox["z", "2"]]]]], "+", RowBox[List[SqrtBox["m"], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", FractionBox["1", "m"], "]"]], "-", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcSin", "[", "z", "]"]], ",", FractionBox["1", "m"]]], "]"]]]], ")"]]]]]], RowBox[List["2", " ", "m", " ", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]]]]], "/;", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "&&", RowBox[List[SuperscriptBox["z", "2"], ">", "1"]], "&&", RowBox[List[RowBox[List[SuperscriptBox["z", "2"], "-", "m"]], ">", "0"]], "&&", RowBox[List["m", "<", "1"]]]]]]]]]]

2001-10-29