(EllipticThetaPrime[1, 0, q] EllipticTheta[1, z, q])/ (4 EllipticTheta[3, 0, q] EllipticTheta[3, z, q]) == Sum[(-1)^(k - 1) (q^(k - 1/2)/(1 + q^(2 k - 1))) Sin[(2 k - 1) z], {k, 1, Infinity}] /; Abs[Im[z]] < (1/2) Im[\[Tau]] && q == E^(I Pi \[Tau])

Cell[BoxData[RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", "0", ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", "z", ",", "q"]], "]"]]]], RowBox[List["4", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "z", ",", "q"]], "]"]]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", FractionBox[SuperscriptBox["q", RowBox[List["k", "-", RowBox[List["1", "/", "2"]]]]], RowBox[List["1", "+", SuperscriptBox["q", RowBox[List[RowBox[List["2", "k"]], "-", "1"]]]]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "-", "1"]], ")"]], " ", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "<", RowBox[List[FractionBox["1", "2"], RowBox[List["Im", "[", "\[Tau]", "]"]]]]]], "\[And]", RowBox[List["q", "\[Equal]", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]"]]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <mrow> <msubsup> <mi> &#977; </mi> <mn> 1 </mn> <mo> &#8242; </mo> </msubsup> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mfrac> <msup> <mi> q </mi> <mrow> <mi> k </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> q </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#964; </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> q </mi> <mo> &#10869; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <apply> <ci> Subscript </ci> <ci> &#977; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 1 </cn> <ci> z </ci> <ci> q </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <cn type='integer'> 0 </cn> <ci> q </ci> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <ci> z </ci> <ci> q </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> q </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> q </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <imaginary /> <ci> &#964; </ci> </apply> </apply> </apply> <apply> <eq /> <ci> q </ci> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#964; </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", "0", ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", "z_", ",", "q_"]], "]"]]]], RowBox[List["4", " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", "q_"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "z_", ",", "q_"]], "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox["q", RowBox[List["k", "-", FractionBox["1", "2"]]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], " ", "z"]], "]"]]]], RowBox[List["1", "+", SuperscriptBox["q", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "<", FractionBox[RowBox[List["Im", "[", "\[Tau]", "]"]], "2"]]], "&&", RowBox[List["q", "\[Equal]", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]"]]]]]]]]]]]]]

2001-10-29