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http://functions.wolfram.com/09.47.03.0004.01
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InverseJacobiSD[-1, m] == (I/Sqrt[m]) EllipticF[I ArcSinh[Sqrt[m]],
(-1 + m)/m] /; m > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List[RowBox[List["-", "1"]], ",", "m"]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[ImaginaryI]", SqrtBox["m"]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", SqrtBox["m"], "]"]]]], ",", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]], "m"]]], "]"]]]]]], "/;", RowBox[List["m", ">", "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> sd </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mtext> </mtext> </mrow> <msqrt> <mi> m </mi> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> m </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> ❘ </mo> <mfrac> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> InverseJacobiSD </ci> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticF </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <ci> m </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["InverseJacobiSD", "[", RowBox[List[RowBox[List["-", "1"]], ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", SqrtBox["m"], "]"]]]], ",", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]], "m"]]], "]"]]]], SqrtBox["m"]], "/;", RowBox[List["m", ">", "0"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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