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variants of this functions
EllipticTheta




Mathematica Notation

Traditional Notation









Elliptic Functions > EllipticTheta[1,z,q] > Representations through equivalent functions > With related functions > Involving theta functions > Involving theta3(z,q)





http://functions.wolfram.com/09.01.27.0004.02









  


  








Input Form





EllipticTheta[1, z, q] == (-I) q^(1/4) E^(I z) EllipticTheta[3, z + (Pi/2) (1 + \[Tau]), q] /; q == E^(I Pi \[Tau])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", "z", ",", "q"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["q", RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["z", "+", RowBox[List[FractionBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Tau]"]], ")"]]]]]], ",", "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> 1 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mroot> <mi> q </mi> <mn> 4 </mn> </mroot> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#964; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <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'> 1 </cn> <ci> z </ci> <ci> q </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <ci> q </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <pi /> <apply> <plus /> <ci> &#964; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <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["1", ",", "z_", ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["q", RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List["z", "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Tau]"]], ")"]]]]]], ",", "q"]], "]"]]]], "/;", RowBox[List["q", "\[Equal]", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]"]]]]]]]]]]]


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





2001-10-29