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http://functions.wolfram.com/09.19.06.0002.01
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WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}] ==
{20 (Pi/(2 Subscript[\[Omega], 1]))^4
(1/15 + 16 Sum[(k^3 q^(2 k))/(1 - q^(2 k)), {k, 1, Infinity}]),
28 (Pi/(2 Subscript[\[Omega], 1]))^6
(2/189 - (16/3) Sum[(k^5 q^(2 k))/(1 - q^(2 k)), {k, 1, Infinity}])} /;
q == Exp[Pi I (Subscript[\[Omega], 3]/Subscript[\[Omega], 1])]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["20", SuperscriptBox[RowBox[List["(", FractionBox["\[Pi]", RowBox[List["2", SubscriptBox["\[Omega]", "1"]]]], ")"]], "4"], RowBox[List["(", RowBox[List[FractionBox["1", "15"], "+", RowBox[List["16", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["k", "3"], " ", SuperscriptBox["q", RowBox[List["2", "k"]]]]], RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", "k"]]]]]]]]]]]], ")"]]]], ",", RowBox[List["28", " ", SuperscriptBox[RowBox[List["(", FractionBox["\[Pi]", RowBox[List["2", SubscriptBox["\[Omega]", "1"]]]], ")"]], "6"], RowBox[List["(", RowBox[List[FractionBox["2", "189"], "-", RowBox[List[FractionBox["16", "3"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["k", "5"], " ", SuperscriptBox["q", RowBox[List["2", "k"]]]]], RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", "k"]]]]]]]]]]]], ")"]]]]]], "}"]]]], "/;", RowBox[List["q", "\[Equal]", RowBox[List["Exp", "[", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", FractionBox[SubscriptBox["\[Omega]", "3"], SubscriptBox["\[Omega]", "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> <mo> { </mo> <mrow> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> } </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mn> 20 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> π </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 15 </mn> </mfrac> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mi> k </mi> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mn> 28 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> π </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 2 </mn> <mn> 189 </mn> </mfrac> <mo> - </mo> <mrow> <mfrac> <mn> 16 </mn> <mn> 3 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mi> k </mi> <mn> 5 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> q </mi> <mo> ⩵ </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mfrac> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <list> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </list> <list> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 15 </cn> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 28 </cn> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <plus /> <cn type='rational'> 2 <sep /> 189 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 16 <sep /> 3 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> k </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </list> </apply> <apply> <eq /> <ci> q </ci> <apply> <exp /> <apply> <times /> <pi /> <imaginaryi /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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