Weierstrass zeta function: Representations through equivalent functions (formula 09.17.27.0006)

WeierstrassZeta

$Mathematica Notation$

Elliptic Functions

WeierstrassZeta[z,{g₂,g₃}]

Representations through equivalent functions

With related functions

Involving theta functions

http://functions.wolfram.com/09.17.27.0006.01

Input Form

WeierstrassZeta[Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}] == (-(Pi^2/(12 Subscript[\[Omega], 1]))) (Derivative[0, 3, 0][EllipticTheta][1, 0, q]/EllipticThetaPrime[1, 0, q])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["\[Pi]", "2"], RowBox[List["12", SubscriptBox["\[Omega]", "1"]]]]]], " ", FractionBox[RowBox[List[SuperscriptBox["EllipticTheta", TagBox[RowBox[List["(", RowBox[List["0", ",", "3", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", ",", "0", ",", "q"]], "]"]], RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", "0", ",", "q"]], "]"]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msub> <mi> η </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mfrac> <mrow> <msubsup> <mi> ϑ </mi> <mn> 1 </mn> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <msubsup> <mi> ϑ </mi> <mn> 1 </mn> <mo> ′ </mo> </msubsup> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> η </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 0 </cn> <ci> q </ci> </apply> <apply> <power /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <apply> <ci> Subscript </ci> <ci> ϑ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 0 </cn> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]_", "1"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List[SuperscriptBox["EllipticTheta", TagBox[RowBox[List["(", RowBox[List["0", ",", "3", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", ",", "0", ",", "q"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["12", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], ")"]], " ", RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", "0", ",", "q"]], "]"]]]]]]]]]]]

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

2001-10-29