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http://functions.wolfram.com/09.20.27.0006.01
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WeierstrassPHalfPeriodValues[{Subscript[g, 2], Subscript[g, 3]}] ==
(1/3) (Pi/(2 Subscript[\[Omega], 1]))^2
{EllipticTheta[3, 0, q]^4 + EllipticTheta[4, 0, q]^4,
EllipticTheta[2, 0, q]^4 - EllipticTheta[4, 0, q]^4,
-EllipticTheta[2, 0, q]^4 - EllipticTheta[3, 0, q]^4}
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Cell[BoxData[RowBox[List[RowBox[List["WeierstrassPHalfPeriodValues", "[", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "3"], SuperscriptBox[RowBox[List["(", FractionBox["\[Pi]", RowBox[List["2", SubscriptBox["\[Omega]", "1"]]]], ")"]], "2"], RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", "q"]], "]"]], "4"], "+", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", "0", ",", "q"]], "]"]], "4"]]], ",", RowBox[List[SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "0", ",", "q"]], "]"]], "4"], "-", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", "0", ",", "q"]], "]"]], "4"]]], ",", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "0", ",", "q"]], "]"]], "4"]]], "-", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", "q"]], "]"]], "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> <mrow> <mo> { </mo> <mrow> <mrow> <mstyle scriptlevel='0'> <msub> <mi> e </mi> <mn> 1 </mn> </msub> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mstyle scriptlevel='0'> <mo> , </mo> </mstyle> <mrow> <mstyle scriptlevel='0'> <msub> <mi> e </mi> <mn> 2 </mn> </msub> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> , </mo> <mrow> <mstyle scriptlevel='0'> <msub> <mi> e </mi> <mn> 3 </mn> </msub> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> <mstyle scriptlevel='0'> <mo> } </mo> </mstyle> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <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> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msup> <mrow> <msub> <mi> ϑ </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> + </mo> <msup> <mrow> <msub> <mi> ϑ </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> , </mo> <mrow> <msup> <mrow> <msub> <mi> ϑ </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> - </mo> <msup> <mrow> <msub> <mi> ϑ </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <msup> <mrow> <msub> <mi> ϑ </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <msup> <mrow> <msub> <mi> ϑ </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <list> <apply> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> </list> <apply> <times /> <cn type='rational'> 1 <sep /> 3 </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'> 2 </cn> </apply> <list> <apply> <plus /> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <cn type='integer'> 0 </cn> <ci> q </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <cn type='integer'> 0 </cn> <ci> q </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 2 </cn> <cn type='integer'> 0 </cn> <ci> q </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <cn type='integer'> 0 </cn> <ci> q </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 2 </cn> <cn type='integer'> 0 </cn> <ci> q </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <cn type='integer'> 0 </cn> <ci> q </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </list> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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