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http://functions.wolfram.com/09.49.27.0003.01
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DedekindEta[z] == E^((Pi I z)/12) EllipticTheta[3, (Pi/2) (z + 1),
E^(3 Pi I z)] /; Im[z] > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DedekindEta", "[", "z", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "z"]], "12"]], RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List[FractionBox[RowBox[List[" ", "\[Pi]"]], "2"], RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["3", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "z"]]]]], "]"]]]]]], "/;", " ", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ">", "0"]]]]]]
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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> <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> <mo> ⩵ </mo> <mrow> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 12 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> ϑ </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <times /> <pi /> <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> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DedekindEta", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "z"]], "12"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["3", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "z"]]]]], "]"]]]], "/;", RowBox[List[RowBox[List["Im", "[", "z", "]"]], ">", "0"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
