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http://functions.wolfram.com/06.06.20.0008.01
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D[Gamma[a, z], {a, \[Alpha]}] ==
Integrate[(t^(a - 1) (a Log[t])^\[Alpha] GammaRegularized[-\[Alpha], 0,
a Log[t]])/E^t, {t, z, Infinity}]/a^\[Alpha]
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Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]]], RowBox[List["Gamma", "[", RowBox[List["a", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["a", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "z", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["a", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", RowBox[List["Log", "[", "t", "]"]]]], ")"]], "\[Alpha]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "t"]]], RowBox[List["GammaRegularized", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", "0", ",", RowBox[List["a", " ", RowBox[List["Log", "[", "t", "]"]]]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> α </mi> </msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> a </mi> <mi> α </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <msup> <mi> a </mi> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mi> z </mi> <mi> ∞ </mi> </msubsup> <mrow> <msup> <mi> t </mi> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> </msup> <mo> ⁢ </mo> <mtext> </mtext> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <semantics> <mi> Q </mi> <annotation-xml encoding='MathML-Content'> <ci> GammaRegularized </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> a </ci> <degree> <ci> α </ci> </degree> </bvar> <apply> <ci> Gamma </ci> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <ci> z </ci> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <ln /> <ci> t </ci> </apply> </apply> <ci> α </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <ci> GammaRegularized </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <cn type='integer'> 0 </cn> <apply> <times /> <ci> a </ci> <apply> <ln /> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["Gamma", "[", RowBox[List["a_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["a", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "z", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t", RowBox[List["a", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", RowBox[List["Log", "[", "t", "]"]]]], ")"]], "\[Alpha]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "t"]]], " ", RowBox[List["GammaRegularized", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", "0", ",", RowBox[List["a", " ", RowBox[List["Log", "[", "t", "]"]]]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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