|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/04.13.23.0017.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Sum[(BernoulliB[n]/n!) z^n, {n, 0, Infinity}] == z/(E^z - 1) /; Abs[z] < 2 Pi
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["BernoulliB", "[", "n", "]"]], RowBox[List["n", "!"]]], " ", SuperscriptBox["z", "n"]]]]], "\[Equal]", FractionBox["z", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], "-", "1"]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", RowBox[List["2", " ", "\[Pi]"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mi> n </mi> </msub> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> ⩵ </mo> <mfrac> <mi> z </mi> <mrow> <msup> <mi> ⅇ </mi> <mi> z </mi> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </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> <mo> < </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> BernoulliB </ci> <ci> n </ci> </apply> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <exponentiale /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["BernoulliB", "[", "n_", "]"]], " ", SuperscriptBox["z_", "n_"]]], RowBox[List["n_", "!"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["z", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], "-", "1"]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", RowBox[List["2", " ", "\[Pi]"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|