D[DedekindEta[z], z] == ((I Pi)/12) DedekindEta[z] + Pi I E^((Pi I z)/12) Sum[(-1)^k k (3 k - 1) E^(Pi I k (3 k - 1) z), {k, -Infinity, Infinity}]

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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> <mfrac> <mrow> <mo> &#8706; </mo> <semantics> <mrow> <mi> &#951; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Eta]&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> </mrow> <mrow> <mo> &#8706; </mo> <mi> z </mi> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 12 </mn> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#951; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Eta]&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 12 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <imaginaryi /> <pi /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> k </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <ci> k </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29