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InverseJacobiCN






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/09.38.21.0002.01









  


  










Input Form





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










Standard Form





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MathML Form







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Rule Form





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










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





2001-10-29





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