Associated Weierstrass sigma function: General characteristics (formula 09.16.04.0008)

WeierstrassSigma

$Mathematica Notation$

Elliptic Functions

WeierstrassSigma[n,z,{g₂,g₃}]

General characteristics

Symmetries and periodicities

Homogeneity

http://functions.wolfram.com/09.16.04.0008.01

Input Form

WeierstrassSigma[n, \[Lambda] z, WeierstrassInvariants[ \[Lambda] WeierstrassHalfPeriods[{Subscript[g, 2], Subscript[g, 3]}]]] == WeierstrassSigma[n, z, {Subscript[g, 2], Subscript[g, 3]}] /; Element[n, {1, 2, 3}]

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> <mrow> <semantics> <mrow> <msub> <mi> σ </mi> <mi> n </mi> </msub> <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[SubscriptBox["\[Sigma]", "n"], "(", 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> <msub> <mi> σ </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ; </mo> <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> </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> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox["\[Sigma]", "n"], "(", RowBox[List[RowBox[List[TagBox["z", Rule[Editable, True]], ";", TagBox[RowBox[List[SubscriptBox["g", "2"], "(", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], ")"]], Rule[Editable, True]]]], ",", TagBox[RowBox[List[SubscriptBox["g", "3"], "(", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], ")"]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mrow> <mo> { </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <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> <ci> z </ci> <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> </apply> </apply> <apply> <in /> <ci> n </ci> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> <cn type='integer'> 3 </cn> </list> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassSigma", "[", RowBox[List["n_", ",", RowBox[List["\[Lambda]_", " ", "z_"]], ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["\[Lambda]_", " ", RowBox[List["WeierstrassHalfPeriods", "[", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]], "]"]]]], "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["WeierstrassSigma", "[", RowBox[List["n", ",", "z", ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]], "/;", RowBox[List["n", "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2", ",", "3"]], "}"]]]]]]]]]]

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

2001-10-29

WeierstrassSigma[z,{g₂,g₃}]