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http://functions.wolfram.com/01.02.24.0003.01
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Product[(1 - q^k)^a, {k, 1, Infinity}] ==
(1/q^(a/24)) DedekindEta[-((I Log[q])/(2 Pi))]^a /; Element[a, Reals]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[Infinity]"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["q", "k"]]], ")"]], "a"]]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox["q", RowBox[List["a", "/", "24"]]]], SuperscriptBox[RowBox[List["DedekindEta", "[", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Log", "[", "q", "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "a"]]]]], "/;", RowBox[List["a", "\[Element]", "Reals"]]]]]]
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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> <mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> q </mi> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> q </mi> <mrow> <mi> a </mi> <mo> / </mo> <mn> 24 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["log", "(", "q", ")"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mi> a </mi> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> a </mi> <mo> ∈ </mo> <mi> ℝ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> q </ci> <ci> k </ci> </apply> </apply> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> q </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 24 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> DedekindEta </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ln /> <ci> q </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> a </ci> </apply> </apply> </apply> <apply> <in /> <ci> a </ci> <ci> ℝ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["q_", "k_"]]], ")"]], "a_"]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["DedekindEta", "[", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Log", "[", "q", "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "a"], SuperscriptBox["q", RowBox[List["a", "/", "24"]]]], "/;", RowBox[List["a", "\[Element]", "Reals"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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