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http://functions.wolfram.com/09.03.03.0002.01
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EllipticTheta[3, (Pi/2) (\[Tau] + 1), E^(3 Pi I \[Tau])] ==
DedekindEta[\[Tau]]/E^((Pi I \[Tau])/12) /; Im[\[Tau]] > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List[FractionBox["\[Pi]", "2"], RowBox[List["(", RowBox[List["\[Tau]", "+", "1"]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["3", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "\[Tau]"]]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Tau]"]], "12"]]]], RowBox[List["DedekindEta", "[", "\[Tau]", "]"]]]]]], "/;", " ", RowBox[List[RowBox[List["Im", "[", "\[Tau]", "]"]], ">", "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> <mrow> <msub> <mi> ϑ </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> τ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> τ </mi> </mrow> <mn> 12 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> τ </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox["\[Tau]", Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> τ </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> τ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> <imaginaryi /> <ci> τ </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <ci> τ </ci> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> DedekindEta </ci> <ci> τ </ci> </apply> </apply> </apply> <apply> <gt /> <apply> <imaginary /> <ci> τ </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["EllipticTheta", "[", RowBox[List["3", ",", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Tau]_", "+", "1"]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["3", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "\[Tau]_"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "12"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Tau]"]], ")"]]]]], " ", RowBox[List["DedekindEta", "[", "\[Tau]", "]"]]]], "/;", RowBox[List[RowBox[List["Im", "[", "\[Tau]", "]"]], ">", "0"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
