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InverseJacobiNC






Mathematica Notation

Traditional Notation









Elliptic Functions > InverseJacobiNC[z,m] > Integration > Indefinite integration > Involving only one direct function with respect to m





http://functions.wolfram.com/09.43.21.0002.01









  


  










Input Form





Integrate[InverseJacobiNC[z, m], m] == -2 ((z - z Sqrt[m + z^2 - m z^2])/Sqrt[z^2 - 1] + Sqrt[1 - m] (EllipticE[I ArcSinh[Sqrt[1 - m]/Sqrt[m]], m/(m - 1)] - EllipticF[I ArcSinh[Sqrt[1 - m]/Sqrt[m]], m/(m - 1)] - EllipticE[I ArcSinh[(Sqrt[1 - m] z)/Sqrt[m]], m/(m - 1)] + EllipticF[I ArcSinh[(Sqrt[1 - m] z)/Sqrt[m]], m/(m - 1)])) /; z > 1 && m > 0










Standard Form





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MathML Form







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</mo> <mrow> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> m </ci> </bvar> <apply> <ci> InverseJacobiNC </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticF </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> EllipticF </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["InverseJacobiNC", "[", RowBox[List["z_", ",", "m_"]], "]"]], RowBox[List["\[DifferentialD]", "m_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["z", "-", RowBox[List["z", " ", SqrtBox[RowBox[List["m", "+", SuperscriptBox["z", "2"], "-", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]]]]]], SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[SqrtBox[RowBox[List["1", "-", "m"]]], SqrtBox["m"]], "]"]]]], ",", FractionBox["m", RowBox[List["m", "-", "1"]]]]], "]"]], "-", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[SqrtBox[RowBox[List["1", "-", "m"]]], SqrtBox["m"]], "]"]]]], ",", FractionBox["m", RowBox[List["m", "-", "1"]]]]], "]"]], "-", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " ", "z"]], SqrtBox["m"]], "]"]]]], ",", FractionBox["m", RowBox[List["m", "-", "1"]]]]], "]"]], "+", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", "m"]]], " ", "z"]], SqrtBox["m"]], "]"]]]], ",", FractionBox["m", RowBox[List["m", "-", "1"]]]]], "]"]]]], ")"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["z", ">", "1"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]










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





2001-10-29





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