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http://functions.wolfram.com/09.49.17.0005.01
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DedekindEta[z + 1/3]^3 == E^((Pi I)/12) DedekindEta[z]^3 +
(3 Sqrt[3] DedekindEta[9 z]^3)/E^((Pi I)/12)
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Cell[BoxData[RowBox[List[SuperscriptBox[RowBox[List["DedekindEta", "[", RowBox[List["z", "+", FractionBox["1", "3"]]], "]"]], "3"], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "12"]], SuperscriptBox[RowBox[List["DedekindEta", "[", "z", "]"]], "3"]]], "+", RowBox[List["3", " ", SqrtBox["3"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "12"]]]], SuperscriptBox[RowBox[List["DedekindEta", "[", RowBox[List["9", "z"]], "]"]], "3"]]]]]]]]]
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<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> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox[RowBox[List["z", "+", FractionBox["1", "3"]]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> <mo> ⩵ </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 12 </mn> </mfrac> </msup> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox["z", Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msqrt> <mn> 3 </mn> </msqrt> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 12 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox[RowBox[List["9", " ", "z"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <power /> <apply> <ci> DedekindEta </ci> <apply> <plus /> <ci> z </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 12 </cn> </apply> <apply> <times /> <pi /> <imaginaryi /> </apply> </apply> </apply> <apply> <power /> <apply> <ci> DedekindEta </ci> <apply> <times /> <cn type='integer'> 9 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["DedekindEta", "[", RowBox[List["z_", "+", FractionBox["1", "3"]]], "]"]], "3"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "12"]], " ", SuperscriptBox[RowBox[List["DedekindEta", "[", "z", "]"]], "3"]]], "+", RowBox[List["3", " ", SqrtBox["3"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "12"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["DedekindEta", "[", RowBox[List["9", " ", "z"]], "]"]], "3"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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