RamanujanTau[n] == (1/2) Integrate[DedekindEta[z]^24/E^(2 Pi I n z), {z, -1 + I \[Gamma], 1 + I \[Gamma]}] /; Element[n, Integers] && n >= 0 && Re[\[Gamma]] > 0

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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> <mi> RamanujanTau </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <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;, Identity, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 24 </mn> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8805; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#947; </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> RamanujanTau </ci> <ci> n </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <int /> <bvar> <ci> z </ci> </bvar> <lowlimit> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> &#947; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> &#947; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <pi /> <imaginaryi /> <ci> n </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> <cn type='integer'> 24 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <geq /> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <ci> &#947; </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02