EllipticTheta[1, Pi (m + 1/2), q] == (-1)^m Sqrt[2/Pi] InverseEllipticNomeQ[q]^(1/4) Sqrt[EllipticK[InverseEllipticNomeQ[q]]] /; Element[m, Integers]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["m", "+", FractionBox["1", "2"]]], ")"]]]], ",", "q"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SqrtBox[FractionBox["2", "\[Pi]"]], SuperscriptBox[RowBox[List["InverseEllipticNomeQ", "[", "q", "]"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "q", "]"]], "]"]]]]]]], "/;", RowBox[List["m", "\[Element]", "Integers"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> &#977; </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mroot> <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> <mn> 4 </mn> </mroot> <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> <mo> /; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 1 </cn> <apply> <times /> <pi /> <apply> <plus /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> q </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </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> InverseEllipticNomeQ </ci> <ci> q </ci> </apply> <cn type='rational'> 1 <sep /> 4 </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> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["m_", "+", FractionBox["1", "2"]]], ")"]]]], ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", SuperscriptBox[RowBox[List["InverseEllipticNomeQ", "[", "q", "]"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox[RowBox[List["EllipticK", "[", RowBox[List["InverseEllipticNomeQ", "[", "q", "]"]], "]"]]]]], "/;", RowBox[List["m", "\[Element]", "Integers"]]]]]]]]

2007-05-02