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variants of this functions
WeierstrassSigma






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassSigma[z,{g2,g3}] > Product representations > Infinite products involving q, trigonometrics and exponentials





http://functions.wolfram.com/09.15.08.0004.01









  


  










Input Form





WeierstrassSigma[z, {Subscript[g, 2], Subscript[g, 3]}] == ((2 Subscript[\[Omega], 1])/Pi) Sin[(Pi z)/(2 Subscript[\[Omega], 1])] Exp[(Subscript[\[Eta], 1] z^2)/(2 Subscript[\[Omega], 1])] Product[(1 - 2 q^(2 n) Cos[(Pi z)/Subscript[\[Omega], 1]] + q^(4 n))/ (1 - q^(2 n))^2, {n, 1, Infinity}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["WeierstrassSigma", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["2", SubscriptBox["\[Omega]", "1"]]], "\[Pi]"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "z"]], RowBox[List["2", SubscriptBox["\[Omega]", "1"]]]], "]"]], " ", RowBox[List["Exp", "[", FractionBox[RowBox[List[SubscriptBox["\[Eta]", "1"], " ", SuperscriptBox["z", "2"]]], RowBox[List["2", SubscriptBox["\[Omega]", "1"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["n", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["1", "-", RowBox[List["2", SuperscriptBox["q", RowBox[List["2", "n"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "z"]], SubscriptBox["\[Omega]", "1"]], "]"]]]], "+", SuperscriptBox["q", RowBox[List["4", "n"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", "n"]]]]], ")"]], "2"]]]]]]]]]]










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> &#963; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Sigma]&quot;, &quot;(&quot;, RowBox[List[RowBox[List[TagBox[&quot;z&quot;, Rule[Editable, True]], &quot;;&quot;, TagBox[SubscriptBox[&quot;g&quot;, &quot;2&quot;], Rule[Editable, True]]]], &quot;,&quot;, TagBox[SubscriptBox[&quot;g&quot;, &quot;3&quot;], Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mi> &#960; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <msub> <mi> &#951; </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mi> q </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> WeierstrassSigma </ci> <ci> z </ci> <list> <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> </list> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <exp /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> z </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassSigma", "[", RowBox[List["z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"]]], ")"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "z"]], RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"]]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SubscriptBox["\[Eta]", "1"], " ", SuperscriptBox["z", "2"]]], RowBox[List["2", " ", SubscriptBox["\[Omega]", "1"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["n", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["q", RowBox[List["2", " ", "n"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "z"]], SubscriptBox["\[Omega]", "1"]], "]"]]]], "+", SuperscriptBox["q", RowBox[List["4", " ", "n"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", " ", "n"]]]]], ")"]], "2"]]]]]], "\[Pi]"]]]]]










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





2001-10-29