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http://functions.wolfram.com/09.44.06.0009.01
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InverseJacobiND[z, m] \[Proportional] (-(I/2)) Log[m/16] (1 + O[m]) +
(Sqrt[1 - z^2]/Sqrt[z^2 - 1]) ArcTanh[z] (1 + O[m])
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Cell[BoxData[RowBox[List[RowBox[List["InverseJacobiND", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", "2"]]], RowBox[List["Log", "[", FractionBox["m", "16"], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "m", "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]], RowBox[List["ArcTanh", "[", "z", "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "m", "]"]]]], ")"]]]]]]]]]]
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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> <msup> <mi> nd </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> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mtext> </mtext> </mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mi> ⅈ </mi> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mn> 16 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> InverseJacobiND </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <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> <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> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctanh /> <ci> z </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 16 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiND", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["Log", "[", FractionBox["m", "16"], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "m", "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["ArcTanh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "m", "]"]]]], ")"]]]], SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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