InverseJacobiDS[z, 1/2] == Sqrt[2] EllipticF[ArcSin[1/(Sqrt[2] z)], -1] /; z > 1

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["InverseJacobiDS", "[", RowBox[List["z", ",", FractionBox["1", "2"]]], "]"]], "\[Equal]", RowBox[List[SqrtBox["2"], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["ArcSin", "[", FractionBox["1", RowBox[List[SqrtBox["2"], " ", "z"]]], "]"]], ",", RowBox[List["-", "1"]]]], "]"]]]]]], "/;", RowBox[List["z", ">", "1"]]]]]]

<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> ds </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10072; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> InverseJacobiDS </ci> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> EllipticF </ci> <apply> <arcsin /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <gt /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiDS", "[", RowBox[List["z_", ",", FractionBox["1", "2"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox["2"], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["ArcSin", "[", FractionBox["1", RowBox[List[SqrtBox["2"], " ", "z"]]], "]"]], ",", RowBox[List["-", "1"]]]], "]"]]]], "/;", RowBox[List["z", ">", "1"]]]]]]]]

2001-10-29