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http://functions.wolfram.com/09.49.07.0001.01
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DedekindEta[z] == (Gamma[1/4]/(2 Pi^(3/4)))
E^((I/Pi) Integrate[WeierstrassZeta[1, WeierstrassInvariants[{1, t}]],
{t, I, z}])
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Cell[BoxData[RowBox[List[RowBox[List["DedekindEta", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Gamma", "[", FractionBox["1", "4"], "]"]], RowBox[List["2", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "4"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["\[ImaginaryI]", "\[Pi]"], RowBox[List[SubsuperscriptBox["\[Integral]", "\[ImaginaryI]", "z"], RowBox[List[RowBox[List["WeierstrassZeta", "[", RowBox[List["1", ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List["1", ",", "t"]], "}"]], "]"]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]]]]]
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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> <semantics> <mrow> <mi> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox["z", Identity, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo>  </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mfrac> <mi> ⅈ </mi> <mi> π </mi> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mi> ⅈ </mi> <mi> z </mi> </msubsup> <mrow> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <semantics> <mn> 1 </mn> <annotation encoding='Mathematica'> TagBox["1", Rule[Editable, True]] </annotation> </semantics> <mo> ; </mo> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <imaginaryi /> </lowlimit> <uplimit> <ci> z </ci> </uplimit> <apply> <ci> Zeta </ci> <apply> <ci> CompoundExpression </ci> <apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> <ci> t </ci> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 1 </cn> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DedekindEta", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["Gamma", "[", FractionBox["1", "4"], "]"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List[SubsuperscriptBox["\[Integral]", "\[ImaginaryI]", "z"], RowBox[List[RowBox[List["WeierstrassZeta", "[", RowBox[List["1", ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List["1", ",", "t"]], "}"]], "]"]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "\[Pi]"]]]], RowBox[List["2", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "4"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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