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variants of this functions
EllipticTheta




Mathematica Notation

Traditional Notation









Elliptic Functions > EllipticTheta[3,z,q] > Transformations > Transformations and argument simplifications > Argument involving basic arithmetic operations





http://functions.wolfram.com/09.03.16.0001.01









  


  








Input Form





EllipticTheta[3, z/\[Tau], E^(-((I Pi)/\[Tau]))] == (Sqrt[\[Tau]]/Sqrt[I]) Exp[(I z^2)/(Pi \[Tau])] EllipticTheta[3, z, q] /; q == E^(I Pi \[Tau])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", FractionBox["z", "\[Tau]"], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "\[Tau]"]]]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[SqrtBox["\[Tau]"], SqrtBox["\[ImaginaryI]"]], " ", RowBox[List["Exp", "[", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], RowBox[List["\[Pi]", " ", "\[Tau]"]]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "z", ",", "q"]], "]"]]]]]], "/;", RowBox[List["q", "\[Equal]", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]"]]]]]]]]]








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> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mfrac> <mi> z </mi> <mi> &#964; </mi> </mfrac> <mo> , </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mi> &#964; </mi> </mfrac> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msqrt> <mi> &#964; </mi> </msqrt> <msqrt> <mi> &#8520; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> q </mi> <mo> &#10869; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> &#964; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <ci> &#964; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> &#964; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <imaginaryi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <exp /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <pi /> <ci> &#964; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <ci> z </ci> <ci> q </ci> </apply> </apply> </apply> <apply> <eq /> <ci> q </ci> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#964; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", FractionBox["z_", "\[Tau]_"], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "\[Tau]_"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox["\[Tau]"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], RowBox[List["\[Pi]", " ", "\[Tau]"]]]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "z", ",", "q"]], "]"]]]], SqrtBox["\[ImaginaryI]"]], "/;", RowBox[List["q", "\[Equal]", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]"]]]]]]]]]]]


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





2001-10-29