Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

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
 











variants of this functions
EllipticThetaPrime






Mathematica Notation

Traditional Notation









Elliptic Functions > EllipticThetaPrime[2,z,q] > Representations through equivalent functions > With related functions > Involving theta functions > Involving theta4(z,q)





http://functions.wolfram.com/09.06.27.0024.01









  


  










Input Form





EllipticThetaPrime[2, z, q] == (q^(1/4 + m + m^2) ((-I) (1 + 2 m) EllipticTheta[4, z - (1/2) (1 + 2 m) Pi (-1 + \[Tau]), q] + EllipticThetaPrime[4, z - (1/2) (1 + 2 m) Pi (-1 + \[Tau]), q]))/E^(I (1 + 2 m) z) /; Element[m, Integers] && q == E^(I Pi \[Tau])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticThetaPrime", "[", RowBox[List["2", ",", "z", ",", "q"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", "z"]]], " ", SuperscriptBox["q", RowBox[List[FractionBox["1", "4"], "+", "m", "+", SuperscriptBox["m", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["z", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Tau]"]], ")"]]]]]], ",", "q"]], "]"]]]], "+", RowBox[List["EllipticThetaPrime", "[", RowBox[List["4", ",", RowBox[List["z", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Tau]"]], ")"]]]]]], ",", "q"]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", 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> <msubsup> <mi> &#977; </mi> <mn> 2 </mn> <mo> &#8242; </mo> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </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> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> q </mi> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msubsup> <mi> &#977; </mi> <mn> 4 </mn> <mo> &#8242; </mo> </msubsup> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <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> <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> <mo> - </mo> <mrow> <mi> &#8520; </mi> <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> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <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> <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> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </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> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <apply> <ci> Subscript </ci> <ci> &#977; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> z </ci> <ci> q </ci> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> q </ci> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <ci> m </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <apply> <plus /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <apply> <ci> Subscript </ci> <ci> &#977; </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <pi /> <apply> <plus /> <ci> &#964; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> q </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <pi /> <apply> <plus /> <ci> &#964; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> q </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <eq /> <ci> q </ci> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#964; </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticThetaPrime", "[", RowBox[List["2", ",", "z_", ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", "z"]]], " ", SuperscriptBox["q", RowBox[List[FractionBox["1", "4"], "+", "m", "+", SuperscriptBox["m", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", RowBox[List["z", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Tau]"]], ")"]]]]]], ",", "q"]], "]"]]]], "+", RowBox[List["EllipticThetaPrime", "[", RowBox[List["4", ",", RowBox[List["z", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]]]], ")"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Tau]"]], ")"]]]]]], ",", "q"]], "]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["q", "\[Equal]", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Tau]"]]]]]]]]]]]]]










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





2007-05-02