JacobiCS[EllipticK[m]/2 + (I EllipticK[1 - m])/2, m] == (1 - I) ((1 - m)^(1/4)/(I Sqrt[1 - Sqrt[m]] + Sqrt[1 + Sqrt[m]]))

Cell[BoxData[RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["EllipticK", "[", "m", "]"]], "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]], "2"]]], ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], FractionBox[RadicalBox[RowBox[List["1", "-", "m"]], "4"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SqrtBox["m"]]]]]], "+", SqrtBox[RowBox[List["1", "+", SqrtBox["m"]]]]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> cs </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mfrac> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mn> 4 </mn> </mroot> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mi> m </mi> </msqrt> </mrow> </msqrt> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mi> m </mi> </msqrt> </mrow> </msqrt> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> JacobiCS </ci> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <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> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='complex-cartesian'> 1 <sep /> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["EllipticK", "[", "m_", "]"]], "2"], "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m_"]], "]"]]]]]], ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["1", "/", "4"]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SqrtBox["m"]]]]]], "+", SqrtBox[RowBox[List["1", "+", SqrtBox["m"]]]]]]]]]]]

2001-10-29