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
 











WeierstrassHalfPeriods






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassHalfPeriods[{g2,g3}] > Representations through equivalent functions > With related functions > Involving elliptic integrals





http://functions.wolfram.com/09.18.27.0003.01









  


  










Input Form





{Subscript[\[Omega], 1], Subscript[\[Omega], 3]} == {EllipticK[m]/Sqrt[Subscript[e, 1] - Subscript[e, 3]], (I EllipticK[1 - m])/Sqrt[Subscript[e, 1] - Subscript[e, 3]]} /; {Subscript[e, 1], Subscript[e, 2], Subscript[e, 3]} == {WeierstrassP[Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}], WeierstrassP[Subscript[\[Omega], 1] + Subscript[\[Omega], 3], {Subscript[g, 2], Subscript[g, 3]}], WeierstrassP[ Subscript[\[Omega], 3], {Subscript[g, 2], Subscript[g, 3]}]} && m == InverseEllipticNomeQ[E^((I Pi Subscript[\[Omega], 3])/ Subscript[\[Omega], 1])] && {Subscript[\[Omega], 1], Subscript[\[Omega], 3]} == WeierstrassHalfPeriods[{Subscript[g, 2], Subscript[g, 3]}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "\[Equal]", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["EllipticK", "[", "m", "]"]], SqrtBox[RowBox[List[SubscriptBox["e", "1"], "-", SubscriptBox["e", "3"]]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]], SqrtBox[RowBox[List[SubscriptBox["e", "1"], "-", SubscriptBox["e", "3"]]]]]]], "}"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["e", "1"], ",", SubscriptBox["e", "2"], ",", SubscriptBox["e", "3"]]], "}"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], ",", RowBox[List["WeierstrassP", "[", RowBox[List[RowBox[List[SubscriptBox["\[Omega]", "1"], "+", SubscriptBox["\[Omega]", "3"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], ",", RowBox[List["WeierstrassP", "[", RowBox[List[SubscriptBox["\[Omega]", "3"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "}"]]]], "\[And]", RowBox[List["m", "\[Equal]", RowBox[List["InverseEllipticNomeQ", "[", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", SubscriptBox["\[Omega]", "3"]]], SubscriptBox["\[Omega]", "1"]]], "]"]]]], "\[And]", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassHalfPeriods", "[", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "]"]]]]]]]]]]










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> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mfrac> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <msqrt> <mrow> <msub> <mi> e </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> e </mi> <mn> 3 </mn> </msub> </mrow> </msqrt> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <msub> <mi> e </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> e </mi> <mn> 3 </mn> </msub> </mrow> </msqrt> </mfrac> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> e </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> e </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> e </mi> <mn> 3 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mi> &#8472; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <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> <mo> , </mo> <mrow> <mi> &#8472; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </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> <mo> , </mo> <mrow> <mi> &#8472; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> <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> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#10869; </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <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> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mstyle scriptlevel='0'> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mstyle scriptlevel='0'> <mo> , </mo> </mstyle> <mrow> <mstyle scriptlevel='0'> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> <mstyle scriptlevel='0'> <mo> } </mo> </mstyle> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <list> <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> </list> <list> <apply> <times /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </list> </apply> <apply> <and /> <apply> <eq /> <list> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 3 </cn> </apply> </list> <list> <apply> <ci> WeierstrassP </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </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> <ci> WeierstrassP </ci> <apply> <plus /> <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> <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> <ci> WeierstrassP </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </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> </list> </apply> <apply> <eq /> <ci> m </ci> <apply> <ci> InverseEllipticNomeQ </ci> <apply> <exp /> <apply> <times /> <imaginaryi /> <pi /> <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> <apply> <eq /> <list> <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> </list> <list> <apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <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> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> <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> </apply> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]_", "1"], ",", SubscriptBox["\[Omega]_", "3"]]], "}"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["EllipticK", "[", "m", "]"]], SqrtBox[RowBox[List[SubscriptBox["e", "1"], "-", SubscriptBox["e", "3"]]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "m"]], "]"]]]], SqrtBox[RowBox[List[SubscriptBox["e", "1"], "-", SubscriptBox["e", "3"]]]]]]], "}"]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["e", "1"], ",", SubscriptBox["e", "2"], ",", SubscriptBox["e", "3"]]], "}"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], ",", RowBox[List["WeierstrassP", "[", RowBox[List[RowBox[List[SubscriptBox["\[Omega]", "1"], "+", SubscriptBox["\[Omega]", "3"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], ",", RowBox[List["WeierstrassP", "[", RowBox[List[SubscriptBox["\[Omega]", "3"], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "}"]]]], "&&", RowBox[List["m", "\[Equal]", RowBox[List["InverseEllipticNomeQ", "[", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", SubscriptBox["\[Omega]", "3"]]], SubscriptBox["\[Omega]", "1"]]], "]"]]]], "&&", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassHalfPeriods", "[", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "]"]]]]]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.