Weierstrass sigma function: General characteristics (formula 09.15.04.0009)

WeierstrassSigma

$Mathematica Notation$

Elliptic Functions

WeierstrassSigma[z,{g₂,g₃}]

General characteristics

Symmetries and periodicities

Homogeneity

http://functions.wolfram.com/09.15.04.0009.01

Input Form

WeierstrassSigma[\[Lambda] z, WeierstrassInvariants[ \[Lambda] WeierstrassHalfPeriods[{Subscript[g, 2], Subscript[g, 3]}]]] == WeierstrassSigma[\[Lambda] z, {Subscript[g, 2]/\[Lambda]^4, Subscript[g, 3]/\[Lambda]^6}]

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> λ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ; </mo> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> λ </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mrow> <mi> λ </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> λ </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mrow> <mi> λ </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox[RowBox[List["\[Lambda]", " ", "z"]], Rule[Editable, True]], ";", TagBox[RowBox[List[SubscriptBox["g", "2"], "(", RowBox[List[RowBox[List["\[Lambda]", " ", SubscriptBox["\[Omega]", "1"]]], ",", RowBox[List["\[Lambda]", " ", SubscriptBox["\[Omega]", "3"]]]]], ")"]], Rule[Editable, True]]]], ",", TagBox[RowBox[List[SubscriptBox["g", "3"], "(", RowBox[List[RowBox[List["\[Lambda]", " ", SubscriptBox["\[Omega]", "1"]]], ",", RowBox[List["\[Lambda]", " ", SubscriptBox["\[Omega]", "3"]]]]], ")"]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> <mo> ⩵ </mo> <semantics> <mrow> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> λ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ; </mo> <mfrac> <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> <msup> <mi> λ </mi> <mn> 4 </mn> </msup> </mfrac> </mrow> <mo> , </mo> <mfrac> <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> <msup> <mi> λ </mi> <mn> 6 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox[RowBox[List["\[Lambda]", " ", "z"]], Rule[Editable, True]], ";", TagBox[FractionBox[RowBox[List[SubscriptBox["g", "2"], "(", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], ")"]], SuperscriptBox["\[Lambda]", "4"]], Rule[Editable, True]]]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["g", "3"], "(", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], ")"]], SuperscriptBox["\[Lambda]", "6"]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> WeierstrassSigma </ci> <apply> <times /> <ci> λ </ci> <ci> z </ci> </apply> <list> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> λ </ci> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> λ </ci> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <ci> λ </ci> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> λ </ci> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </list> </apply> <apply> <ci> WeierstrassSigma </ci> <apply> <times /> <ci> λ </ci> <ci> z </ci> </apply> <list> <apply> <times /> <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> <power /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <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> <apply> <power /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </list> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29

WeierstrassSigma[n,z,{g₂,g₃}]