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InverseJacobiNS






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/09.45.27.0018.01









  


  










Input Form





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










Standard Form





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MathML Form







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</mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mi> tan </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mi> m </mi> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> InverseJacobiNS </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> JacobiCD </ci> <apply> <ci> InverseJacobiNS </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> EllipticF </ci> <apply> <arccsc /> <ci> z </ci> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <not /> <apply> <exists /> <bvar> <ci> &#964; </ci> </bvar> <bvar> <list> <apply> <in /> <ci> &#964; </ci> <reals /> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> &#964; </ci> <cn type='integer'> 1 </cn> </apply> </list> </bvar> <apply> <and /> <apply> <eq /> <apply> <imaginary /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <tan /> <apply> <times /> <pi /> <ci> &#964; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <tan /> <apply> <times /> <pi /> <ci> &#964; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <imaginary /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <tan /> <apply> <times /> <pi /> <ci> &#964; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiNS", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["JacobiCD", "[", RowBox[List[RowBox[List["InverseJacobiNS", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["m", SuperscriptBox["z", "2"]]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["ArcCsc", "[", "z", "]"]], ",", "m"]], "]"]]]], RowBox[List[RowBox[List["-", "1"]], "+", 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"], "-", "1"]], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List["Tan", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Tau]"]], "2"], "]"]]]], ")"]], "2"], "-", "1"]], "<", "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"]], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List["Tan", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Tau]"]], "2"], "]"]]]], ")"]], "2"], "-", "m"]], "<", "0"]]]], ")"]]]]]]]]]]]]










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





2007-05-02





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