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

 

Developed with Mathematica -- Download a Free Trial Version
 











DedekindEta




Mathematica Notation

Traditional Notation









Elliptic Functions > DedekindEta[z] > Series representations > Other series representations





http://functions.wolfram.com/09.49.06.0003.01









  


  








Input Form





DedekindEta[z] == Sum[(KroneckerDelta[1, Mod[k, 12]] - KroneckerDelta[5, Mod[k, 12]] - KroneckerDelta[7, Mod[k, 12]] + KroneckerDelta[11, Mod[k, 12]]) E^((1/12) Pi I k^2 z), {k, 1, Infinity}]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["DedekindEta", "[", "z", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["1", ",", RowBox[List["Mod", "[", RowBox[List["k", " ", ",", "12"]], "]"]]]], "]"]], "-", RowBox[List["KroneckerDelta", "[", RowBox[List["5", ",", RowBox[List["Mod", "[", RowBox[List["k", " ", ",", "12"]], "]"]]]], "]"]], "-", RowBox[List["KroneckerDelta", "[", RowBox[List["7", ",", RowBox[List["Mod", "[", RowBox[List["k", " ", ",", "12"]], "]"]]]], "]"]], "+", RowBox[List["KroneckerDelta", "[", RowBox[List["11", ",", RowBox[List["Mod", "[", RowBox[List["k", " ", ",", "12"]], "]"]]]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "12"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", SuperscriptBox["k", "2"], " ", "z"]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mi> &#951; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Eta]&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mn> 1 </mn> <mo> , </mo> <semantics> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 12 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> FE`Conversion`Private`k </ci> <cn type='integer'> 12 </cn> </apply> </annotation-xml> </semantics> </mrow> </msub> <mo> - </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mn> 5 </mn> <mo> , </mo> <semantics> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 12 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> FE`Conversion`Private`k </ci> <cn type='integer'> 12 </cn> </apply> </annotation-xml> </semantics> </mrow> </msub> <mo> - </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mn> 7 </mn> <mo> , </mo> <semantics> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 12 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> FE`Conversion`Private`k </ci> <cn type='integer'> 12 </cn> </apply> </annotation-xml> </semantics> </mrow> </msub> <mo> + </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mn> 11 </mn> <mo> , </mo> <semantics> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 12 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> FE`Conversion`Private`k </ci> <cn type='integer'> 12 </cn> </apply> </annotation-xml> </semantics> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 12 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <apply> <ci> KroneckerDelta </ci> <cn type='integer'> 1 </cn> <apply> <rem /> <ci> FE`Conversion`Private`k </ci> <cn type='integer'> 12 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> KroneckerDelta </ci> <cn type='integer'> 5 </cn> <apply> <rem /> <ci> FE`Conversion`Private`k </ci> <cn type='integer'> 12 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> KroneckerDelta </ci> <cn type='integer'> 7 </cn> <apply> <rem /> <ci> FE`Conversion`Private`k </ci> <cn type='integer'> 12 </cn> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <cn type='integer'> 11 </cn> <apply> <rem /> <ci> FE`Conversion`Private`k </ci> <cn type='integer'> 12 </cn> </apply> </apply> </apply> <apply> <exp /> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DedekindEta", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["1", ",", RowBox[List["Mod", "[", RowBox[List["k", ",", "12"]], "]"]]]], "]"]], "-", RowBox[List["KroneckerDelta", "[", RowBox[List["5", ",", RowBox[List["Mod", "[", RowBox[List["k", ",", "12"]], "]"]]]], "]"]], "-", RowBox[List["KroneckerDelta", "[", RowBox[List["7", ",", RowBox[List["Mod", "[", RowBox[List["k", ",", "12"]], "]"]]]], "]"]], "+", RowBox[List["KroneckerDelta", "[", RowBox[List["11", ",", RowBox[List["Mod", "[", RowBox[List["k", ",", "12"]], "]"]]]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "12"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", SuperscriptBox["k", "2"], " ", "z"]]]]]]]]]]]


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





2001-10-29