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variants of this functions
Factorial




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Factorial[n] > Differentiation > Fractional integro-differentiation





http://functions.wolfram.com/06.01.20.0005.01









  


  








Input Form





D[n!, {n, \[Alpha]}] == Integrate[(t^n (n Log[t])^\[Alpha] (1 - GammaRegularized[-\[Alpha], n Log[t]]))/E^t, {t, 1, Infinity}]/n^\[Alpha] - Sum[((-1)^k/k!) HypergeometricPFQRegularized[{1, 1}, {1 - \[Alpha]}, -(n/k)], {k, 1, Infinity}]/n^\[Alpha]








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["n_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["n_", "!"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["n", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "1", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t", "n"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", " ", RowBox[List["Log", "[", "t", "]"]]]], ")"]], "\[Alpha]"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["GammaRegularized", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", RowBox[List["n", " ", RowBox[List["Log", "[", "t", "]"]]]]]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "t"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "-", RowBox[List[SuperscriptBox["n", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Alpha]"]], "}"]], ",", RowBox[List["-", FractionBox["n", "k"]]]]], "]"]]]], RowBox[List["k", "!"]]]]]]]]]]]]]


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





2001-10-29