Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Factorial




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Factorial[n] > Series representations > Asymptotic series expansions





http://functions.wolfram.com/06.01.06.0009.01









  


  








Input Form





n! \[Proportional] (Sqrt[2 Pi] n^(n + 1/2) (1 + Sum[((-1)^j PermutationCyclesD[2 (k + j), j])/(2^(k + j) (k + j)!)/ n^k, {k, 1, Infinity}, {j, 1, 2 k}]))/E^n /; (Abs[Arg[n]] < Pi && (Abs[n] -> Infinity) && PermutationCyclesD[m, j] == (m - 1) (PermutationCyclesD[m - 1, j] + (m - 2) PermutationCyclesD[m - 3, j - 1]) && PermutationCyclesD[0, 0] == 1 && PermutationCyclesD[m, 1] == (m - 1)! && PermutationCyclesD[m, j] == 0 /; m <= 3 j - 1)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["n", "!"]], "\[Proportional]", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], SuperscriptBox["n", RowBox[List["n", "+", FractionBox["1", "2"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", "n"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["2", " ", "k"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["PermutationCyclesD", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "j"]], ")"]]]], ",", "j"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["k", "+", "j"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "j"]], ")"]], "!"]]]], ")"]]]], SuperscriptBox["n", RowBox[List["-", "k"]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "n", "]"]], "]"]], "<", "\[Pi]"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List["m", ",", "j"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List[RowBox[List["m", "-", "1"]], ",", "j"]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "2"]], ")"]], RowBox[List["PermutationCyclesD", "[", RowBox[List[RowBox[List["m", "-", "3"]], ",", RowBox[List["j", "-", "1"]]]], "]"]]]]]], ")"]]]]]], "\[And]", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List["0", ",", "0"]], "]"]], "\[Equal]", "1"]], "\[And]", "\[IndentingNewLine]", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List["m", ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]]], "\[And]", "\[IndentingNewLine]", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List["m", ",", "j"]], "]"]], "\[Equal]", "0"]]]], "/;", " ", RowBox[List["m", "\[LessEqual]", RowBox[List[RowBox[List["3", "j"]], "-", "1"]]]]]], ")"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> &#8733; </mo> <mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msup> <mi> n </mi> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> P </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> n </mi> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mi> &#960; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> n </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> P </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> P </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> , </mo> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> P </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> P </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> P </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> P </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> &#8804; </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <factorial /> <ci> n </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <ci> P </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <arg /> <ci> n </ci> </apply> </apply> <pi /> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> n </ci> </apply> <infinity /> </apply> <apply> <eq /> <apply> <ci> P </ci> <ci> m </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> <apply> <ci> P </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -3 </cn> </apply> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> P </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> P </ci> <cn type='integer'> 0 </cn> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> P </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> P </ci> <ci> m </ci> <ci> j </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <leq /> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["n_", "!"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SuperscriptBox["n", RowBox[List["n", "+", FractionBox["1", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["2", " ", "k"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["PermutationCyclesD", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "j"]], ")"]]]], ",", "j"]], "]"]]]], ")"]], " ", SuperscriptBox["n", RowBox[List["-", "k"]]]]], RowBox[List[SuperscriptBox["2", RowBox[List["k", "+", "j"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "j"]], ")"]], "!"]]]]]]]]]]], ")"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "n", "]"]], "]"]], "<", "\[Pi]"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List["m", ",", "j"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List[RowBox[List["m", "-", "1"]], ",", "j"]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "2"]], ")"]], " ", RowBox[List["PermutationCyclesD", "[", RowBox[List[RowBox[List["m", "-", "3"]], ",", RowBox[List["j", "-", "1"]]]], "]"]]]]]], ")"]]]]]], "&&", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List["0", ",", "0"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List["m", ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]]], "&&", RowBox[List[RowBox[List["PermutationCyclesD", "[", RowBox[List["m", ",", "j"]], "]"]], "\[Equal]", "0"]]]], "/;", RowBox[List["m", "\[LessEqual]", RowBox[List[RowBox[List["3", " ", "j"]], "-", "1"]]]]]], ")"]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29