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DedekindEta




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/09.49.06.0013.01









  


  








Input Form





DedekindEta[z]^3 == (Pi^(3/2)/(-Log[E^(Pi I z)])^(3/2)) E^(Pi^2/(4 Log[E^(Pi I z)])) Sum[(-1)^k E^((k (k + 1) Pi^2)/Log[E^(Pi I z)]) (2 k + 1), {k, 0, Infinity}]








Standard Form





Cell[BoxData[RowBox[List[SuperscriptBox[RowBox[List["DedekindEta", "[", "z", "]"]], "3"], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List["Log", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "z"]]], "]"]]]], ")"]], RowBox[List["3", "/", "2"]]]], SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["\[Pi]", "2"], RowBox[List["4", " ", RowBox[List["Log", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "z"]]], "]"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], SuperscriptBox["\[Pi]", "2"]]], RowBox[List["Log", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "z"]]], "]"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msup> <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;, Identity, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> <mo> &#63449; </mo> <mrow> <mfrac> <msup> <mi> &#960; </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> ) </mo> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ln /> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> k </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["DedekindEta", "[", "z_", "]"]], "3"], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["\[Pi]", "2"], RowBox[List["4", " ", RowBox[List["Log", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "z"]]], "]"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"]]], RowBox[List["Log", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "z"]]], "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List["Log", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "z"]]], "]"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]]]


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





2007-05-02