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http://functions.wolfram.com/09.41.21.0002.01
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Integrate[InverseJacobiDN[z, m], m] ==
2 Sqrt[m - 1] (EllipticE[1/(1 - m)] - EllipticE[ArcSin[z], 1/(1 - m)]) /;
z < 1 && m > 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["InverseJacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["\[DifferentialD]", "m"]]]]]], "\[Equal]", RowBox[List["2", " ", SqrtBox[RowBox[List["m", "-", "1"]]], RowBox[List["(", " ", RowBox[List[RowBox[List["EllipticE", "[", FractionBox["1", RowBox[List["1", "-", "m"]]], "]"]], "-", " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcSin", "[", "z", "]"]], ",", FractionBox["1", RowBox[List["1", "-", "m"]]]]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["z", "<", "1"]], "\[And]", RowBox[List["m", ">", "1"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> dn </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> <mo> ⅆ </mo> <mi> m </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> E </mi> <mo> ⁡ </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> ❘ </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> < </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> > </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> m </ci> </bvar> <apply> <ci> InverseJacobiDN </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <ci> EllipticE </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <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 /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["InverseJacobiDN", "[", RowBox[List["z_", ",", "m_"]], "]"]], RowBox[List["\[DifferentialD]", "m_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["m", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", FractionBox["1", RowBox[List["1", "-", "m"]]], "]"]], "-", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcSin", "[", "z", "]"]], ",", FractionBox["1", RowBox[List["1", "-", "m"]]]]], "]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["z", "<", "1"]], "&&", RowBox[List["m", ">", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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