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WeierstrassInvariants




Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassInvariants[{w1,w3}] > Primary definition





http://functions.wolfram.com/09.19.02.0001.02









  


  








Input Form





WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}] == {60 Sum[If[{m, n} == {0, 0}, 0, 1/(2 m Subscript[\[Omega], 1] + 2 n Subscript[\[Omega], 3])^4], {m, -Infinity, Infinity}, {n, -Infinity, Infinity}], 140 Sum[If[{m, n} == {0, 0}, 0, 1/(2 m Subscript[\[Omega], 1] + 2 n Subscript[\[Omega], 3])^6], {m, -Infinity, Infinity}, {n, -Infinity, 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["60", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List["If", "[", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["m", ",", "n"]], "}"]], "\[Equal]", RowBox[List["{", RowBox[List["0", ",", "0"]], "}"]]]], ",", "0", ",", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", "n", " ", SubscriptBox["\[Omega]", "3"]]]]], ")"]], "4"]]]], "]"]]]]]]]], ",", RowBox[List["140", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List["If", "[", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["m", ",", "n"]], "}"]], "\[Equal]", RowBox[List["{", RowBox[List["0", ",", "0"]], "}"]]]], ",", "0", ",", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["2", "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> <mn> 60 </mn> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <munder> <mrow> <mi> m </mi> <mo> , </mo> <mtext> </mtext> <mrow> <mi> n </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> </mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8800; </mo> <mrow> <mo> { </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> } </mo> </mrow> </mrow> </munder> <mi> &#8734; </mi> </munderover> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <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> </mrow> <mo> , </mo> <mrow> <mn> 140 </mn> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <munder> <mrow> <mi> m </mi> <mo> , </mo> <mtext> </mtext> <mrow> <mi> n </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> </mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8800; </mo> <mrow> <mo> { </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> } </mo> </mrow> </mrow> </munder> <mi> &#8734; </mi> </munderover> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <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> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#969; </ms> <ms> 1 </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> &#969; </ms> <ms> 3 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> g </ms> <ms> 3 </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#969; </ms> <ms> 1 </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> &#969; </ms> <ms> 3 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> } </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <ms> 60 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> ErrorBox </ci> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> UnderscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> = </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> &#8734; </ms> </list> </apply> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> , </ms> <ms> n </ms> </list> </apply> <ms> } </ms> </list> </apply> <ms> &#8800; </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <ms> 0 </ms> <ms> , </ms> <ms> 0 </ms> </list> </apply> <ms> } </ms> </list> </apply> </list> </apply> </apply> <ms> &#8734; </ms> </apply> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> m </ms> <apply> <ci> SubscriptBox </ci> <ms> &#969; </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> <apply> <ci> SubscriptBox </ci> <ms> &#969; </ms> <ms> 3 </ms> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 4 </ms> </apply> </apply> </list> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> 140 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> UnderscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> = </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> &#8734; </ms> </list> </apply> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> , </ms> <ms> n </ms> </list> </apply> <ms> } </ms> </list> </apply> <ms> &#8800; </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <ms> 0 </ms> <ms> , </ms> <ms> 0 </ms> </list> </apply> <ms> } </ms> </list> </apply> </list> </apply> </apply> <ms> &#8734; </ms> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> m </ms> <apply> <ci> SubscriptBox </ci> <ms> &#969; </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> n </ms> <apply> <ci> SubscriptBox </ci> <ms> &#969; </ms> <ms> 3 </ms> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 6 </ms> </apply> </apply> </list> </apply> </list> </apply> <ms> } </ms> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Im </ms> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SubscriptBox </ci> <ms> &#969; </ms> <ms> 3 </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> &#969; </ms> <ms> 1 </ms> </apply> </apply> <ms> ) </ms> </list> </apply> <ms> &#8800; </ms> <ms> 0 </ms> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </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["60", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List["If", "[", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["m", ",", "n"]], "}"]], "\[Equal]", RowBox[List["{", RowBox[List["0", ",", "0"]], "}"]]]], ",", "0", ",", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "+", RowBox[List["2", " ", "n", " ", SubscriptBox["\[Omega]\[Omega]", "3"]]]]], ")"]], "4"]]]], "]"]]]]]]]], ",", RowBox[List["140", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List["If", "[", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["m", ",", "n"]], "}"]], "\[Equal]", RowBox[List["{", RowBox[List["0", ",", "0"]], "}"]]]], ",", "0", ",", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]], "+", RowBox[List["2", " ", "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