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InverseJacobiDS






Mathematica Notation

Traditional Notation









Elliptic Functions > InverseJacobiDS[z,m] > Representations through equivalent functions > With related functions > Involving elliptic integrals





http://functions.wolfram.com/09.42.27.0017.01









  


  










Input Form





InverseJacobiDS[z, m] == (-((I z^2 JacobiNC[InverseJacobiDS[z, m], m])/ (Sqrt[m] (m + z^2)))) Sqrt[(z^2 + m)/z^2] Sqrt[(-1 + m + z^2)/z^2] EllipticF[I ArcSinh[Sqrt[m]/z], 1 - 1/m] /; !Exists[\[Tau], {Element[\[Tau], Reals], 0 < \[Tau] < 1}, Im[(z + Tan[(Pi \[Tau])/2])^2 + m] == 0 && (z + Tan[(Pi \[Tau])/2])^2 + m < 0 && Im[(z + Tan[(Pi \[Tau])/2])^2 + m - 1] == 0 && (z + Tan[(Pi \[Tau])/2])^2 + m - 1 < 0]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiDS", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"], " ", RowBox[List["JacobiNC", "[", RowBox[List[RowBox[List["InverseJacobiDS", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[SuperscriptBox["z", "2"], "+", "m"]], SuperscriptBox["z", "2"]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", "m", "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "2"]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox[SqrtBox["m"], "z"], "]"]]]], ",", RowBox[List["1", "-", FractionBox["1", "m"]]]]], "]"]]]], RowBox[List[SqrtBox["m"], " ", RowBox[List["(", RowBox[List["m", "+", SuperscriptBox["z", "2"]]], ")"]]]]]]], "/;", RowBox[List["!", RowBox[List[SubscriptBox["\[Exists]", RowBox[List["\[Tau]", ",", RowBox[List["{", RowBox[List[RowBox[List["\[Tau]", "\[Element]", "Reals"]], ",", RowBox[List["0", "<", "\[Tau]", "<", "1"]]]], "}"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List["Tan", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Tau]"]], "2"], "]"]]]], ")"]], "2"], "+", "m"]], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List["Tan", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Tau]"]], "2"], "]"]]]], ")"]], "2"], "+", "m"]], "<", "0"]], "&&", RowBox[List[RowBox[List["Im", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List["Tan", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Tau]"]], "2"], "]"]]]], ")"]], "2"], "+", "m", "-", "1"]], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List["Tan", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Tau]"]], "2"], "]"]]]], ")"]], "2"], "+", "m", "-", "1"]], "<", "0"]]]], ")"]]]]]]]]]]]]










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





2007-05-02