EllipticTheta[3, z, q] \[Proportional] (((I Sqrt[Pi])/Sqrt[q - 1]) (1 + (1/4) (q - 1) - (7/96) (q - 1)^2 + \[Ellipsis]) E^(z^2/Log[q]) (1 + 2 E^(Pi^2/Log[q]) Cosh[(2 Pi z)/Log[q]] + 2 E^((4 Pi^2)/Log[q]) Cosh[(4 Pi z)/Log[q]] + \[Ellipsis]))/E^(I Pi Floor[3/4 - Arg[q - 1]/(2 Pi)]) /; (q -> 1) && Abs[q] < 1

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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> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> </mrow> <msqrt> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 96 </mn> </mfrac> <mo> &#8290; </mo> <mn> 7 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> q </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <ci> z </ci> <ci> q </ci> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> <apply> <floor /> <apply> <plus /> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 96 </cn> <cn type='integer'> 7 </cn> <apply> <power /> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> z </ci> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> <ci> z </ci> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <apply> <abs /> <ci> q </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02