|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/04.23.06.0006.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
BellB[n] \[Proportional]
(n^n E^(-n + (n Log[Log[n]]^2)/(2 Log[n]^2) + (n (1 + Log[Log[n]]))/
Log[n]))/Log[n]^n /; (n -> Infinity)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BellB", "[", "n", "]"]], "\[Proportional]", RowBox[List[SuperscriptBox["n", "n"], " ", SuperscriptBox[RowBox[List["Log", "[", "n", "]"]], RowBox[List["-", "n"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "n"]], "+", FractionBox[RowBox[List["n", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]], "2"]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["Log", "[", "n", "]"]], "2"]]]], "+", FractionBox[RowBox[List["n", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]]]], ")"]]]], RowBox[List["Log", "[", "n", "]"]]]]]]]]]], "/;", RowBox[List["(", RowBox[List["n", "\[Rule]", "\[Infinity]"]], ")"]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BellB] </annotation> </semantics> <mi> n </mi> </msub> <mo> ∝ </mo> <mrow> <msup> <mi> n </mi> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mfrac> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> BellB </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <power /> <ci> n </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <ln /> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <ln /> <apply> <ln /> <ci> n </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ln /> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <ci> n </ci> <apply> <plus /> <apply> <ln /> <apply> <ln /> <ci> n </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> n </ci> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BellB", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["n", "n"], " ", SuperscriptBox[RowBox[List["Log", "[", "n", "]"]], RowBox[List["-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "n"]], "+", FractionBox[RowBox[List["n", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]], "2"]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["Log", "[", "n", "]"]], "2"]]]], "+", FractionBox[RowBox[List["n", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]]]], ")"]]]], RowBox[List["Log", "[", "n", "]"]]]]]]]], "/;", RowBox[List["(", RowBox[List["n", "\[Rule]", "\[Infinity]"]], ")"]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|