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http://functions.wolfram.com/09.43.06.0001.02
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InverseJacobiNC[z, m] \[Proportional] I EllipticK[1 - m] +
(Sqrt[1 - z^2]/(Sqrt[m] Sqrt[z^2 - 1])) (z + ((2 m - 1) z^3)/(6 m) +
((3 - 8 m + 8 m^2) z^5)/(40 m^2) + \[Ellipsis]) /; (z -> 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["InverseJacobiNC", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " "]], RowBox[List[SqrtBox["m"], " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]], RowBox[List["(", RowBox[List["z", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "-", "1"]], ")"]], " ", SuperscriptBox["z", "3"]]], RowBox[List["6", " ", "m"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "-", RowBox[List["8", " ", "m"]], "+", RowBox[List["8", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", SuperscriptBox["z", "5"]]], RowBox[List["40", " ", SuperscriptBox["m", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]
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<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> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mrow> <msqrt> <mi> m </mi> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mrow> <mn> 40 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mo> … </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> InverseJacobiNC </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <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> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 40 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <ci> … </ci> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiNC", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List["z", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "-", "1"]], ")"]], " ", SuperscriptBox["z", "3"]]], RowBox[List["6", " ", "m"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "-", RowBox[List["8", " ", "m"]], "+", RowBox[List["8", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", SuperscriptBox["z", "5"]]], RowBox[List["40", " ", SuperscriptBox["m", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List[SqrtBox["m"], " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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