D[Subfactorial[z], {z, \[Alpha]}] == Integrate[((t - 1)^z (z Log[t - 1])^\[Alpha] GammaRegularized[-\[Alpha], 0, z Log[t - 1]])/E^t, {t, 0, Infinity}]/z^\[Alpha] /; Re[z] > -1

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["Subfactorial", "[", "z", "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["t", "-", "1"]], ")"]], "z"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", " ", RowBox[List["Log", "[", RowBox[List["t", "-", "1"]], "]"]]]], ")"]], "\[Alpha]"], RowBox[List["GammaRegularized", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", "0", ",", RowBox[List["z", " ", RowBox[List["Log", "[", RowBox[List["t", "-", "1"]], "]"]]]]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "t"]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", RowBox[List["-", "1"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> &#945; </mi> </msup> <mrow> <mi> Subfactorial </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> &#945; </mi> </msup> </mrow> </mfrac> <mo> &#63449; </mo> <mrow> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> z </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> Q </mi> <annotation-xml encoding='MathML-Content'> <ci> GammaRegularized </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> &#945; </ci> </degree> </bvar> <apply> <ci> Subfactorial </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <ln /> <apply> <plus /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> &#945; </ci> </apply> <apply> <ci> GammaRegularized </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <ci> z </ci> <apply> <ln /> <apply> <plus /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["Subfactorial", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["t", "-", "1"]], ")"]], "z"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", " ", RowBox[List["Log", "[", RowBox[List["t", "-", "1"]], "]"]]]], ")"]], "\[Alpha]"], " ", RowBox[List["GammaRegularized", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", "0", ",", RowBox[List["z", " ", RowBox[List["Log", "[", RowBox[List["t", "-", "1"]], "]"]]]]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "t"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", RowBox[List["-", "1"]]]]]]]]]]

2007-05-02