Weierstrass invariants: General characteristics (formula 09.19.04.0006)

WeierstrassInvariants

$Mathematica Notation$

Elliptic Functions

WeierstrassInvariants[{w₁,w₃}]

General characteristics

Symmetries and periodicities

Homogeneity

http://functions.wolfram.com/09.19.04.0006.01

Input Form

WeierstrassInvariants[{\[Lambda] Subscript[\[Omega], 1], \[Lambda] Subscript[\[Omega], 3]}] == {\[Lambda]^(-4), \[Lambda]^(-6)} WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}] /; Element[\[Lambda], Complexes]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[RowBox[List["\[Lambda]", " ", SubscriptBox["\[Omega]", "1"]]], ",", RowBox[List["\[Lambda]", " ", SubscriptBox["\[Omega]", "3"]]]]], "}"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["{", RowBox[List[SuperscriptBox["\[Lambda]", RowBox[List["-", "4"]]], ",", SuperscriptBox["\[Lambda]", RowBox[List["-", "6"]]]]], "}"]], " ", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]]]]]], "/;", RowBox[List["\[Lambda]", "\[Element]", "Complexes"]]]]]]

MathML Form

<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> <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> <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> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> λ </mi> <mn> 4 </mn> </msup> </mfrac> <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> <mfrac> <mn> 1 </mn> <msup> <mi> λ </mi> <mn> 6 </mn> </msup> </mfrac> <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> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> λ </mi> <mo> ∈ </mo> <semantics> <mi> ℂ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalC]", Function[Complexes]] </annotation> </semantics> </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> <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> <list> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <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> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> λ </ci> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </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> </apply> </list> </apply> <apply> <in /> <ci> λ </ci> <complexes /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[RowBox[List["\[Lambda]_", " ", SubscriptBox["\[Omega]_", "1"]]], ",", RowBox[List["\[Lambda]_", " ", SubscriptBox["\[Omega]_", "3"]]]]], "}"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", SuperscriptBox["\[Lambda]", "4"]], ",", FractionBox["1", SuperscriptBox["\[Lambda]", "6"]]]], "}"]], " ", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]\[Omega]", "1"], ",", SubscriptBox["\[Omega]\[Omega]", "3"]]], "}"]], "]"]]]], "/;", RowBox[List["\[Lambda]", "\[Element]", "Complexes"]]]]]]]]

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

2001-10-29