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 GammaRegularized

 http://functions.wolfram.com/06.08.06.0015.01

 Input Form

 GammaRegularized[-n + \[Epsilon], z] \[Proportional] (-1)^n n! Gamma[-n, z] \[Epsilon] + (-1)^n n! (Gamma[-n, z] (EulerGamma - HarmonicNumber[n] + Log[z]) + MeijerG[{{}, {1, 1}}, {{0, 0, -n}, {}}, z]) \[Epsilon]^2 + (-1)^n n! (1/6) (6 (EulerGamma - HarmonicNumber[n] + Log[z]) MeijerG[{{}, {1, 1}}, {{0, 0, -n}, {}}, z] + 6 MeijerG[{{}, {1, 1, 1}}, {{0, 0, 0, -n}, {}}, z] + Gamma[-n, z] (-Pi^2 + 3 Log[z]^2 + 3 PolyGamma[0, 1 + n] (-2 Log[z] + PolyGamma[0, 1 + n]) + 3 PolyGamma[1, 1 + n])) \[Epsilon]^3 + O[\[Epsilon]^4] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GammaRegularized", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", "\[Epsilon]"]], ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["n", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], "\[Epsilon]"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "-", RowBox[List["HarmonicNumber", "[", "n", "]"]], "+", RowBox[List["Log", "[", "z", "]"]]]], ")"]]]], "+", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0", ",", RowBox[List["-", "n"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], ")"]], SuperscriptBox["\[Epsilon]", "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["(", RowBox[List["EulerGamma", "-", RowBox[List["HarmonicNumber", "[", "n", "]"]], "+", RowBox[List["Log", "[", "z", "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0", ",", RowBox[List["-", "n"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List["6", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", "1", ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0", ",", "0", ",", RowBox[List["-", "n"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "2"]]], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["0", ",", RowBox[List["1", "+", "n"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["0", ",", RowBox[List["1", "+", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "n"]]]], "]"]]]]]], ")"]]]]]], ")"]], SuperscriptBox["\[Epsilon]", "3"]]], "+", RowBox[List["O", "[", SuperscriptBox["\[Epsilon]", "4"], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 Q GammaRegularized ( ϵ - n , z ) ( - 1 ) n n ! Γ ( - n , z ) ϵ + ( - 1 ) n n ! ( Γ ( - n , z ) ( - H HarmonicNumber n + log ( z ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) + G 2 , 3 3 , 0 ( z 1 , 1 0 , 0 , - n ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "3"]], RowBox[List["3", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["1", MeijerG, Rule[Editable, True]], ",", TagBox["1", MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "n"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) ϵ 2 + 1 6 ( - 1 ) n n ! ( 6 ( - H HarmonicNumber n + log ( z ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) G 2 , 3 3 , 0 ( z 1 , 1 0 , 0 , - n ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "3"]], RowBox[List["3", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["1", MeijerG, Rule[Editable, True]], ",", TagBox["1", MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "n"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + 6 G 3 , 4 4 , 0 ( z 1 , 1 , 1 0 , 0 , 0 , - n ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "4"]], RowBox[List["4", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["1", MeijerG, Rule[Editable, True]], ",", TagBox["1", MeijerG, Rule[Editable, True]], ",", TagBox["1", MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "n"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + Γ ( - n , z ) ( 3 log 2 ( z ) - π 2 + 3 ψ TagBox["\[Psi]", PolyGamma] ( 0 ) ( n + 1 ) ( ψ TagBox["\[Psi]", PolyGamma] ( 0 ) ( n + 1 ) - 2 log ( z ) ) + 3 ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( n + 1 ) ) ) ϵ 3 + O ( ϵ 4 ) /; n Condition Proportional GammaRegularized ϵ -1 n z -1 n n Gamma -1 n z ϵ -1 n n Gamma -1 n z -1 HarmonicNumber n z MeijerG 1 1 0 0 -1 n z ϵ 2 1 6 -1 n n 6 -1 HarmonicNumber n z MeijerG 1 1 0 0 -1 n z 6 MeijerG 1 1 1 0 0 0 -1 n z Gamma -1 n z 3 z 2 -1 2 3 PolyGamma 0 n 1 PolyGamma 0 n 1 -1 2 z 3 PolyGamma 1 n 1 ϵ 3 O ϵ 4 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GammaRegularized", "[", RowBox[List[RowBox[List[RowBox[List["-", "n_"]], "+", "\[Epsilon]_"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], " ", "\[Epsilon]"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "-", RowBox[List["HarmonicNumber", "[", "n", "]"]], "+", RowBox[List["Log", "[", "z", "]"]]]], ")"]]]], "+", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0", ",", RowBox[List["-", "n"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "2"]]], "+", RowBox[List[FractionBox["1", "6"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["(", RowBox[List["EulerGamma", "-", RowBox[List["HarmonicNumber", "[", "n", "]"]], "+", RowBox[List["Log", "[", "z", "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0", ",", RowBox[List["-", "n"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List["6", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", "1", ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0", ",", "0", ",", RowBox[List["-", "n"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "2"]]], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["0", ",", RowBox[List["1", "+", "n"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["0", ",", RowBox[List["1", "+", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "n"]]]], "]"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "3"]]], "+", SuperscriptBox[RowBox[List["O", "[", "\[Epsilon]", "]"]], "4"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02