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WeierstrassInvariants




Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassInvariants[{w1,w3}] > Representations through equivalent functions > With related functions > Involving elliptic integrals and modular functions





http://functions.wolfram.com/09.19.27.0012.01









  


  








Input Form





WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}] == {(4 (m^2 - m + 1) EllipticK[m]^4)/(3 Subscript[\[Omega], 1]^4), (4 (m - 2) (2 m - 1) (m + 1) EllipticK[m]^6)/ (27 Subscript[\[Omega], 1]^6)} /; m == ModularLambda[Subscript[\[Omega], 3]/Subscript[\[Omega], 1]]








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[FractionBox[RowBox[List["4", RowBox[List["(", RowBox[List[SuperscriptBox["m", "2"], "-", "m", "+", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "4"]]], RowBox[List["3", SubsuperscriptBox["\[Omega]", "1", "4"]]]], ",", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["m", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", "m"]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "6"]]], RowBox[List["27", SubsuperscriptBox["\[Omega]", "1", "6"]]]]]], "}"]]]], "/;", RowBox[List["m", "\[Equal]", RowBox[List["ModularLambda", "[", FractionBox[SubscriptBox["\[Omega]", "3"], SubscriptBox["\[Omega]", "1"]], "]"]]]]]]]]








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> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msubsup> <mi> &#969; </mi> <mn> 1 </mn> <mn> 4 </mn> </msubsup> </mrow> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> </mrow> <mrow> <mn> 27 </mn> <mo> &#8290; </mo> <msubsup> <mi> &#969; </mi> <mn> 1 </mn> <mn> 6 </mn> </msubsup> </mrow> </mfrac> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> &#10869; </mo> <semantics> <mrow> <mi> &#955; </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> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[FractionBox[SubscriptBox[&quot;\[Omega]&quot;, &quot;3&quot;], SubscriptBox[&quot;\[Omega]&quot;, &quot;1&quot;]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </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> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </list> </apply> <apply> <eq /> <ci> m </ci> <apply> <ci> ModularLambda </ci> <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> </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[FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["m", "2"], "-", "m", "+", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "4"]]], RowBox[List["3", " ", SubsuperscriptBox["\[Omega]\[Omega]", "1", "4"]]]], ",", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["m", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", "m", "]"]], "6"]]], RowBox[List["27", " ", SubsuperscriptBox["\[Omega]\[Omega]", "1", "6"]]]]]], "}"]], "/;", RowBox[List["m", "\[Equal]", RowBox[List["ModularLambda", "[", FractionBox[SubscriptBox["\[Omega]", "3"], SubscriptBox["\[Omega]\[Omega]", "1"]], "]"]]]]]]]]]]


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





2001-10-29