|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/09.49.21.0001.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Integrate[DedekindEta[z], z] == (-((12 I)/Pi))
Sum[((-1)^k (E^(((Pi I)/12) (6 k - 1)^2 z) - 1))/(6 k - 1)^2,
{k, -Infinity, Infinity}]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["DedekindEta", "[", "z", "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["12", "\[ImaginaryI]"]], "\[Pi]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "12"], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["6", " ", "k"]], "-", "1"]], ")"]], "2"], " ", "z"]]], "-", "1"]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["6", " ", "k"]], "-", "1"]], ")"]], "2"]]]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <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> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mi> π </mi> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 12 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <exp /> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["DedekindEta", "[", "z_", "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["12", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "12"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["6", " ", "k"]], "-", "1"]], ")"]], "2"], " ", "z"]]], "-", "1"]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["6", " ", "k"]], "-", "1"]], ")"]], "2"]]]]]], "\[Pi]"]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|