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WeierstrassInvariants






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassInvariants[{w1,w3}] > General characteristics > Symmetries and periodicities > Transformation of half-periods





http://functions.wolfram.com/09.19.04.0005.01









  


  










Input Form





WeierstrassInvariants[{\[Alpha] Subscript[\[Omega], 1] + \[Beta] Subscript[\[Omega], 3], \[Gamma] Subscript[\[Omega], 1] + \[Delta] Subscript[\[Omega], 3]}] == WeierstrassInvariants[ {Subscript[\[Omega], 1], Subscript[\[Omega], 3]}] /; Element[{\[Alpha], \[Beta], \[Gamma], \[Delta]}, Integers] && \[Alpha] \[Delta] - \[Beta] \[Gamma] == \[PlusMinus]1










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["\[Alpha]", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["\[Beta]", " ", SubscriptBox["\[Omega]", "3"]]]]], ",", RowBox[List[RowBox[List["\[Gamma]", " ", SubscriptBox["\[Omega]", "1"]]], "+", RowBox[List["\[Delta]", " ", SubscriptBox["\[Omega]", "3"]]]]]]], "}"]], "]"]], "\[Equal]", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["\[Alpha]", ",", "\[Beta]", ",", "\[Gamma]", ",", "\[Delta]"]], "}"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["\[Alpha]", " ", "\[Delta]"]], "-", RowBox[List["\[Beta]", " ", "\[Gamma]"]]]], "\[Equal]", RowBox[List["\[PlusMinus]", "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> <mrow> <mrow> <mi> &#945; </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mi> &#946; </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mi> &#947; </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mi> &#948; </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#945; </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mi> &#946; </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mi> &#947; </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mi> &#948; </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> } </mo> </mrow> <mo> &#10869; </mo> <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> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> &#945; </mi> <mo> , </mo> <mi> &#946; </mi> <mo> , </mo> <mi> &#947; </mi> <mo> , </mo> <mi> &#948; </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mi> &#945; </mi> <mo> &#8290; </mo> <mi> &#948; </mi> </mrow> <mo> - </mo> <mrow> <mi> &#946; </mi> <mo> &#8290; </mo> <mi> &#947; </mi> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mo> &#177; </mo> <mn> 1 </mn> </mrow> </mrow> </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> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> &#946; </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> &#947; </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> &#948; </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> &#946; </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> &#947; </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> &#948; </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </list> <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> </apply> <apply> <and /> <apply> <in /> <list> <ci> &#945; </ci> <ci> &#946; </ci> <ci> &#947; </ci> <ci> &#948; </ci> </list> <integers /> </apply> <apply> <eq /> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <ci> &#948; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#946; </ci> <ci> &#947; </ci> </apply> </apply> </apply> <apply> <ci> PlusMinus </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["\[Alpha]_", " ", SubscriptBox["\[Omega]_", "1"]]], "+", RowBox[List["\[Beta]_", " ", SubscriptBox["\[Omega]_", "3"]]]]], ",", RowBox[List[RowBox[List["\[Gamma]_", " ", SubscriptBox["\[Omega]_", "1"]]], "+", RowBox[List["\[Delta]_", " ", SubscriptBox["\[Omega]_", "3"]]]]]]], "}"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]\[Omega]", "1"], ",", SubscriptBox["\[Omega]\[Omega]", "3"]]], "}"]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["\[Alpha]", ",", "\[Beta]", ",", "\[Gamma]", ",", "\[Delta]"]], "}"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["\[Alpha]", " ", "\[Delta]"]], "-", RowBox[List["\[Beta]", " ", "\[Gamma]"]]]], "\[Equal]", RowBox[List["\[PlusMinus]", "1"]]]]]]]]]]]]










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





2001-10-29