EllipticThetaPrime[1, (Pi - I Log[q])/2, q] == ((-I) Sqrt[2/Pi] Sqrt[EllipticK[InverseEllipticNomeQ[q]]])/q^4^(-1)

Cell[BoxData[RowBox[List[RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Log", "[", "q", "]"]]]]]], "2"], " ", ",", "q"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["q", RowBox[List[RowBox[List["-", "1"]], "/", "4"]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], SqrtBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "q", "]"]], "]"]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mi> &#977; </mi> <mn> 1 </mn> <mo> &#8242; </mo> </msubsup> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mfrac> <mn> 1 </mn> <mroot> <mi> q </mi> <mn> 4 </mn> </mroot> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <apply> <ci> Subscript </ci> <ci> &#977; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ln /> <ci> q </ci> </apply> </apply> </apply> </apply> </apply> <ci> q </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> q </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> InverseEllipticNomeQ </ci> <ci> q </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Log", "[", "q_", "]"]]]]]], ")"]]]], ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["q", RowBox[List[RowBox[List["-", "1"]], "/", "4"]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", SqrtBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "q", "]"]], "]"]]]]]]]]]

2007-05-02