ModularLambda[z] == 1 - (1 + 2 Sum[(-1)^k E^(k^2 I Pi z), {k, 1, Infinity}])^ 4/(1 + 2 Sum[E^(k^2 I Pi z), {k, 1, Infinity}])^4

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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> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Identity, Rule[Editable, True], Rule[Selectable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#63449; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </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> <msup> <mi> &#8519; </mi> <mrow> <msup> <mi> k </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <msup> <mi> &#8519; </mi> <mrow> <msup> <mi> k </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02