InverseJacobiNC[z, m] == ((Sqrt[-1 + z^2] JacobiDS[InverseJacobiNC[z, m], m])/ Sqrt[m + z^2 - m z^2]) Integrate[1/(Sqrt[t^2 - 1] Sqrt[(1 - m) t^2 + m]), {t, 1, z}] /; !Exists[\[Tau], {Element[\[Tau], Reals], 0 < \[Tau] < 1}, Im[(1 + \[Tau] (z - 1))^2 - 1] == 0 && (1 + \[Tau] (z - 1))^2 - 1 < 0 && Im[(1 - m) (1 + \[Tau] (z - 1))^2 + m] == 0 && (1 - m) (1 + \[Tau] (z - 1))^2 + m < 0]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["InverseJacobiNC", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["InverseJacobiNC", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], SqrtBox[RowBox[List["m", "+", SuperscriptBox["z", "2"], "-", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]]], RowBox[List[SubsuperscriptBox["\[Integral]", "1", "z"], RowBox[List[FractionBox["1", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["t", "2"], "-", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox["t", "2"]]], "+", "m"]]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", " ", RowBox[List["Not", "[", RowBox[List["Exists", "[", RowBox[List["\[Tau]", ",", " ", RowBox[List["{", RowBox[List[RowBox[List["\[Tau]", "\[Element]", "Reals"]], ",", " ", RowBox[List["0", "<", "\[Tau]", "<", "1"]]]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["Im", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Tau]", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]], ")"]], "2"], "-", "1"]], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Tau]", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]], ")"]], "2"], "-", "1"]], "<", "0"]], "\[And]", RowBox[List[RowBox[List["Im", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Tau]", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]], ")"]], "2"]]], "+", "m"]], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Tau]", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]], ")"]], "2"]]], "+", "m"]], "<", "0"]]]]]], "]"]], "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> nc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> ds </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> nc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mi> m </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 1 </mn> <mi> z </mi> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mrow> <msup> <mi> t </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> m </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> &#172; </mo> <mrow> <msub> <mo> &#8707; </mo> <mrow> <mi> &#964; </mi> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mi> &#964; </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> &#964; </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <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> InverseJacobiNC </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <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> <apply> <ci> JacobiDS </ci> <apply> <ci> InverseJacobiNC </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> m </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> z </ci> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> m </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </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 /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <ci> &#964; </ci> </apply> <cn type='integer'> 1 </cn> </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 /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <ci> &#964; </ci> </apply> <cn type='integer'> 1 </cn> </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> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <ci> &#964; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> m </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <ci> &#964; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiNC", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["InverseJacobiNC", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "1", "z"], RowBox[List[FractionBox["1", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["t", "2"], "-", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox["t", "2"]]], "+", "m"]]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], SqrtBox[RowBox[List["m", "+", SuperscriptBox["z", "2"], "-", 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["1", "+", RowBox[List["\[Tau]", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]], ")"]], "2"], "-", "1"]], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Tau]", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]], ")"]], "2"], "-", "1"]], "<", "0"]], "&&", RowBox[List[RowBox[List["Im", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Tau]", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]], ")"]], "2"]]], "+", "m"]], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Tau]", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]], ")"]], "2"]]], "+", "m"]], "<", "0"]]]], ")"]]]]]]]]]]]]

2007-05-02