EllipticThetaPrime[3, m Pi + n I Log[q], q] == (2 n I Sqrt[2/Pi] Sqrt[EllipticK[InverseEllipticNomeQ[q]]])/q^n^2 /; Element[{m, n}, Integers]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mi> &#977; </mi> <mn> 3 </mn> <mo> &#8242; </mo> </msubsup> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <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> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> q </mi> <mrow> <mo> - </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mrow> </msup> <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> <mo> /; </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> } </mo> </mrow> <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> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <apply> <ci> Subscript </ci> <ci> &#977; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> m </ci> <pi /> </apply> <apply> <times /> <ci> n </ci> <imaginaryi /> <apply> <ln /> <ci> q </ci> </apply> </apply> </apply> <ci> q </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> <imaginaryi /> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </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> <apply> <in /> <list> <ci> m </ci> <ci> n </ci> </list> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02