Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

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
 











WeierstrassInvariants






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassInvariants[{w1,w3}] > Series representations > Other series representations





http://functions.wolfram.com/09.19.06.0003.01









  


  










Input Form





WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}] == {(Pi^4/12) (1/Subscript[\[Omega], 1]^4 + 1/Subscript[\[Omega], 3]^4) + (15/2) Sum[1/(m Subscript[\[Omega], 1] - n Subscript[\[Omega], 3])^4 + 1/(m Subscript[\[Omega], 1] + n Subscript[\[Omega], 3])^4, {m, 1, Infinity}, {n, 1, Infinity}], (Pi^6/216) (1/Subscript[\[Omega], 1]^6 + 1/Subscript[\[Omega], 3]^6) + (35/8) Sum[1/(m Subscript[\[Omega], 1] - n Subscript[\[Omega], 3])^6 + 1/(m Subscript[\[Omega], 1] + n Subscript[\[Omega], 3])^6, {m, 1, Infinity}, {n, 1, Infinity}]} /; Im[Subscript[\[Omega], 3]/Subscript[\[Omega], 1]] != 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "4"], "12"], RowBox[List["(", RowBox[List[FractionBox["1", SubsuperscriptBox["\[Omega]", "1", "4"]], "+", FractionBox["1", SubsuperscriptBox["\[Omega]", "3", "4"]]]], ")"]]]], "+", RowBox[List[FractionBox["15", "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], "\[Infinity]"], RowBox[List["(", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Omega]", "1"]]], "-", RowBox[List["n", " ", SubscriptBox["\[Omega]", "3"]]]]], ")"]], "4"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["n", " ", SubscriptBox["\[Omega]", "3"]]]]], ")"]], "4"]]]], ")"]]]]]]]]]], ",", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "6"], "216"], RowBox[List["(", RowBox[List[FractionBox["1", SubsuperscriptBox["\[Omega]", "1", "6"]], "+", FractionBox["1", SubsuperscriptBox["\[Omega]", "3", "6"]]]], ")"]]]], "+", RowBox[List[FractionBox["35", "8"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], "\[Infinity]"], RowBox[List["(", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Omega]", "1"]]], "-", RowBox[List["n", " ", SubscriptBox["\[Omega]", "3"]]]]], ")"]], "6"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["n", " ", SubscriptBox["\[Omega]", "3"]]]]], ")"]], "6"]]]], ")"]]]]]]]]]]]], "}"]]]], "/;", RowBox[List[RowBox[List["Im", "[", FractionBox[SubscriptBox["\[Omega]", "3"], SubscriptBox["\[Omega]", "1"]], "]"]], "\[NotEqual]", "0"]]]]]]










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> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> } </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mrow> <mfrac> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> <mn> 12 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <msubsup> <mi> &#969; </mi> <mn> 3 </mn> <mn> 4 </mn> </msubsup> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msubsup> <mi> &#969; </mi> <mn> 1 </mn> <mn> 4 </mn> </msubsup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 15 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mfrac> <msup> <mi> &#960; </mi> <mn> 6 </mn> </msup> <mn> 216 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <msubsup> <mi> &#969; </mi> <mn> 3 </mn> <mn> 6 </mn> </msubsup> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msubsup> <mi> &#969; </mi> <mn> 1 </mn> <mn> 6 </mn> </msubsup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 35 </mn> <mn> 8 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mfrac> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> &#8800; </mo> <mn> 0 </mn> </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> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </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> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </list> <list> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 15 <sep /> 2 </cn> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <cn type='integer'> 216 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 35 <sep /> 8 </cn> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </list> </apply> <apply> <neq /> <apply> <times /> <imaginary /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]_", "1"], ",", SubscriptBox["\[Omega]_", "3"]]], "}"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "12"], " ", SuperscriptBox["\[Pi]", "4"], " ", RowBox[List["(", RowBox[List[FractionBox["1", SubsuperscriptBox["\[Omega]\[Omega]", "1", "4"]], "+", FractionBox["1", SubsuperscriptBox["\[Omega]\[Omega]", "3", "4"]]]], ")"]]]], "+", RowBox[List[FractionBox["15", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], "\[Infinity]"], RowBox[List["(", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "-", RowBox[List["n", " ", SubscriptBox["\[Omega]\[Omega]", "3"]]]]], ")"]], "4"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "+", RowBox[List["n", " ", SubscriptBox["\[Omega]\[Omega]", "3"]]]]], ")"]], "4"]]]], ")"]]]]]]]]]], ",", RowBox[List[RowBox[List[FractionBox["1", "216"], " ", SuperscriptBox["\[Pi]", "6"], " ", RowBox[List["(", RowBox[List[FractionBox["1", SubsuperscriptBox["\[Omega]\[Omega]", "1", "6"]], "+", FractionBox["1", SubsuperscriptBox["\[Omega]\[Omega]", "3", "6"]]]], ")"]]]], "+", RowBox[List[FractionBox["35", "8"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], "\[Infinity]"], RowBox[List["(", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "-", RowBox[List["n", " ", SubscriptBox["\[Omega]\[Omega]", "3"]]]]], ")"]], "6"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["m", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "+", RowBox[List["n", " ", SubscriptBox["\[Omega]\[Omega]", "3"]]]]], ")"]], "6"]]]], ")"]]]]]]]]]]]], "}"]], "/;", RowBox[List[RowBox[List["Im", "[", FractionBox[SubscriptBox["\[Omega]\[Omega]", "3"], SubscriptBox["\[Omega]\[Omega]", "1"]], "]"]], "\[NotEqual]", "0"]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.