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DedekindEta






Mathematica Notation

Traditional Notation









Elliptic Functions > DedekindEta[z] > Identities > Functional identities





http://functions.wolfram.com/09.49.17.0006.01









  


  










Input Form





DedekindEta[z]^48 == (1/256) (2 DedekindEta[(z + 1)/2]^24 DedekindEta[z/2]^24 + DedekindEta[(z + 1)/2]^16 (E^((2 Pi I)/3) DedekindEta[(z + 1)/2]^16 + DedekindEta[z/2]^16/ E^((2 Pi I)/3)) DedekindEta[z/2]^16)










Standard Form





Cell[BoxData[RowBox[List[SuperscriptBox[RowBox[List["DedekindEta", "[", "z", "]"]], "48"], "\[Equal]", RowBox[List[FractionBox["1", "256"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["DedekindEta", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], "24"], " ", SuperscriptBox[RowBox[List["DedekindEta", "[", FractionBox["z", "2"], "]"]], "24"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["DedekindEta", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], "16"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "3"]], " ", SuperscriptBox[RowBox[List["DedekindEta", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], "16"]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "3"]]]], " ", SuperscriptBox[RowBox[List["DedekindEta", "[", FractionBox["z", "2"], "]"]], "16"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["DedekindEta", "[", FractionBox["z", "2"], "]"]], "16"]]]]], ")"]]]]]]]]










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;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 48 </mn> </msup> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 256 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#951; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Eta]&quot;, &quot;(&quot;, TagBox[FractionBox[RowBox[List[&quot;z&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 24 </mn> </msup> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#951; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Eta]&quot;, &quot;(&quot;, TagBox[FractionBox[&quot;z&quot;, &quot;2&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 24 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mrow> <mi> &#951; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Eta]&quot;, &quot;(&quot;, TagBox[FractionBox[RowBox[List[&quot;z&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 16 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 3 </mn> </mfrac> </msup> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#951; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Eta]&quot;, &quot;(&quot;, TagBox[FractionBox[RowBox[List[&quot;z&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 16 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 3 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#951; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Eta]&quot;, &quot;(&quot;, TagBox[FractionBox[&quot;z&quot;, &quot;2&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 16 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#951; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Eta]&quot;, &quot;(&quot;, TagBox[FractionBox[&quot;z&quot;, &quot;2&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 16 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> <cn type='integer'> 48 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 256 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> DedekindEta </ci> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 24 </cn> </apply> <apply> <power /> <apply> <ci> DedekindEta </ci> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 24 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> DedekindEta </ci> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 16 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> DedekindEta </ci> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 16 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> DedekindEta </ci> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 16 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> DedekindEta </ci> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 16 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["DedekindEta", "[", "z_", "]"]], "48"], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "256"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["DedekindEta", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], "24"], " ", SuperscriptBox[RowBox[List["DedekindEta", "[", FractionBox["z", "2"], "]"]], "24"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["DedekindEta", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], "16"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "3"]], " ", SuperscriptBox[RowBox[List["DedekindEta", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], "16"]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], " ", RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["DedekindEta", "[", FractionBox["z", "2"], "]"]], "16"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["DedekindEta", "[", FractionBox["z", "2"], "]"]], "16"]]]]], ")"]]]]]]]]










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





2001-10-29