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http://functions.wolfram.com/09.49.06.0011.01
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DedekindEta[z] \[Proportional] ((1/Sqrt[(-I) z]) (1 + O[E^(-((2 I Pi)/z))]))/
E^((I Pi)/(12 z)) /; (Abs[z] -> Infinity) && Arg[z] == Pi/2
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DedekindEta", "[", "z", "]"]], "\[Proportional]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], RowBox[List["12", " ", "z"]]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]"]], "z"]]]], "]"]]]], ")"]]]]]], "/;", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List[RowBox[List["Arg", "[", "z", "]"]], "\[Equal]", FractionBox["\[Pi]", "2"]]]]]]]]]
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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", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mo> ∝ </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mi> z </mi> </mfrac> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> <apply> <eq /> <apply> <arg /> <ci> z </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </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]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], RowBox[List["12", " ", "z"]]]]]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["O", "[", "\[ExponentialE]", "]"]], RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]"]], "z"]]]]]], ")"]]]], SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List["Arg", "[", "z", "]"]], "\[Equal]", FractionBox["\[Pi]", "2"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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