Product[1 - q^k, {k, 1, Infinity}] == (1/q^(1/24)) DedekindEta[-((I Log[q])/(2 Pi))]

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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> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> q </mi> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mroot> <mi> q </mi> <mn> 24 </mn> </mroot> </mfrac> <mo> &#8290; </mo> <mrow> <mi> &#951; </mi> <mo> &#8289; </mo> <mo> ( </mo> <semantics> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;-&quot;, FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;q&quot;, &quot;)&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;\[Pi]&quot;]]]]], Rule[Editable, True]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> q </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> q </ci> <cn type='rational'> 1 <sep /> 24 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> &#951; </ci> <apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ln /> <ci> q </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29