Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
WeierstrassSigma




Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassSigma[n,z,{g2,g3}] > General characteristics > Symmetries and periodicities > Periodicity





http://functions.wolfram.com/09.16.04.0006.01









  


  








Input Form





WeierstrassSigma[i, z + 2 m Subscript[\[Omega], i] + 2 n Subscript[\[Omega], j] + 2 r Subscript[\[Omega], k], {Subscript[g, 2], Subscript[g, 3]}] == (-1)^(n r + r m + m n + m) Exp[2 (m Subscript[\[Eta], i] + n Subscript[\[Eta], j] + r Subscript[\[Eta], k]) (z + m Subscript[\[Omega], i] + n Subscript[\[Omega], j] + r Subscript[\[Omega], k])] WeierstrassSigma[i, z, {Subscript[g, 2], Subscript[g, 3]}] /; Element[{i, j, k}, {1, 2, 3}] && i != j != k && Element[{m, n, r}, Integers]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WeierstrassSigma", "[", RowBox[List["i", ",", RowBox[List["z", "+", RowBox[List["2", "m", " ", SubscriptBox["\[Omega]", "i"]]], "+", RowBox[List["2", "n", " ", SubscriptBox["\[Omega]", "j"]]], "+", RowBox[List["2", "r", " ", SubscriptBox["\[Omega]", "k"]]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["n", " ", "r"]], "+", RowBox[List["r", " ", "m"]], "+", RowBox[List["m", " ", "n"]], " ", "+", "m"]]], " ", RowBox[List["Exp", "[", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Eta]", "i"]]], "+", RowBox[List["n", " ", SubscriptBox["\[Eta]", "j"]]], "+", RowBox[List["r", " ", SubscriptBox["\[Eta]", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List["m", " ", SubscriptBox["\[Omega]", "i"]]], "+", RowBox[List["n", " ", SubscriptBox["\[Omega]", "j"]]], "+", RowBox[List["r", " ", SubscriptBox["\[Omega]", "k"]]]]], ")"]]]], "]"]], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List["i", ",", "z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["i", ",", "j", ",", "k"]], "}"]], "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2", ",", "3"]], "}"]]]], "\[And]", RowBox[List["i", "\[NotEqual]", "j", "\[NotEqual]", "k"]], "\[And]", RowBox[List[RowBox[List["{", RowBox[List["m", ",", "n", ",", "r"]], "}"]], "\[Element]", "Integers"]]]]]]]]








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> &#963; </mi> <mi> i </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mi> i </mi> </msub> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mi> j </mi> </msub> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> r </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mi> k </mi> </msub> </mrow> </mrow> <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[SubscriptBox[&quot;\[Sigma]&quot;, &quot;i&quot;], &quot;(&quot;, RowBox[List[RowBox[List[TagBox[RowBox[List[&quot;z&quot;, &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;m&quot;, &quot; &quot;, SubscriptBox[&quot;\[Omega]&quot;, &quot;i&quot;]]], &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;n&quot;, &quot; &quot;, SubscriptBox[&quot;\[Omega]&quot;, &quot;j&quot;]]], &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;r&quot;, &quot; &quot;, SubscriptBox[&quot;\[Omega]&quot;, &quot;k&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> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> r </mi> </mrow> <mo> + </mo> <mrow> <mi> r </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mi> m </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msub> <mi> &#951; </mi> <mi> i </mi> </msub> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msub> <mi> &#951; </mi> <mi> j </mi> </msub> </mrow> <mo> + </mo> <mrow> <mi> r </mi> <mo> &#8290; </mo> <msub> <mi> &#951; </mi> <mi> k </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mi> i </mi> </msub> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mi> j </mi> </msub> </mrow> <mo> + </mo> <mrow> <mi> r </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mi> k </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <msub> <mi> &#963; </mi> <mi> i </mi> </msub> <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[SubscriptBox[&quot;\[Sigma]&quot;, &quot;i&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> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8712; </mo> <mrow> <mo> { </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> i </mi> <mo> &#8800; </mo> <mi> j </mi> <mo> &#8800; </mo> <mi> k </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> <mo> , </mo> <mi> r </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> WeierstrassSigma </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <ci> i </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <ci> j </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <ci> k </ci> </apply> </apply> </apply> <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> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <ci> n </ci> <ci> r </ci> </apply> <apply> <times /> <ci> r </ci> <ci> m </ci> </apply> <apply> <times /> <ci> m </ci> <ci> n </ci> </apply> <ci> m </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <ci> i </ci> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <ci> j </ci> </apply> </apply> <apply> <times /> <ci> r </ci> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <ci> i </ci> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <ci> j </ci> </apply> </apply> <apply> <times /> <ci> r </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <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> </apply> <apply> <and /> <apply> <in /> <list> <ci> i </ci> <ci> j </ci> <ci> k </ci> </list> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> <cn type='integer'> 3 </cn> </list> </apply> <apply> <neq /> <ci> i </ci> <ci> j </ci> <ci> k </ci> </apply> <apply> <in /> <list> <ci> m </ci> <ci> n </ci> <ci> r </ci> </list> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassSigma", "[", RowBox[List["i_", ",", RowBox[List["z_", "+", RowBox[List["2", " ", "m_", " ", SubscriptBox["\[Omega]_", "i_"]]], "+", RowBox[List["2", " ", "n_", " ", SubscriptBox["\[Omega]_", "j"]]], "+", RowBox[List["2", " ", "r_", " ", SubscriptBox["\[Omega]_", "k"]]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["n", " ", "r"]], "+", RowBox[List["r", " ", "m"]], "+", RowBox[List["m", " ", "n"]], "+", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Eta]", "i"]]], "+", RowBox[List["n", " ", SubscriptBox["\[Eta]", "j"]]], "+", RowBox[List["r", " ", SubscriptBox["\[Eta]", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List["m", " ", SubscriptBox["\[Omega]\[Omega]", "i"]]], "+", RowBox[List["n", " ", SubscriptBox["\[Omega]", "j"]]], "+", RowBox[List["r", " ", SubscriptBox["\[Omega]", "k"]]]]], ")"]]]]], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List["i", ",", "z", ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["i", ",", "j", ",", "k"]], "}"]], "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2", ",", "3"]], "}"]]]], "&&", RowBox[List["i", "\[NotEqual]", "j", "\[NotEqual]", "k"]], "&&", RowBox[List[RowBox[List["{", RowBox[List["m", ",", "n", ",", "r"]], "}"]], "\[Element]", "Integers"]]]]]]]]]]


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





2001-10-29