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 Gamma

 http://functions.wolfram.com/06.05.06.0032.01

 Input Form

 Gamma[-n + \[Epsilon]] \[Proportional] ((-1)^n/(n! \[Epsilon])) Sum[((Subscript[p, k] \[Epsilon]^q)/(q - k)!) (1 + Sum[Sum[(Binomial[n, j] (-1)^(j - 1))/j^i, {j, 1, n}] \[Epsilon]^i, {i, 1, Infinity}]), {q, 0, Infinity}, {k, 0, q}] /; Element[n, Integers] && n >= 0 && Subscript[s, 1] == EulerGamma && Subscript[s, k] == Zeta[k] /; k > 1 && Subscript[a, 0] == -EulerGamma && Subscript[a, k] == ((-1)^(k + 1)/(k + 1)) Subscript[s, k + 1] /; k > 0 && Subscript[p, 0] == (-1)^m EulerGamma^m && Subscript[p, k] == (1/(Subscript[a, 0] k)) Sum[(m j - k + j) Subscript[a, j] Subscript[p, k - j], {j, 1, k}] /; k > 0 && m = q - k

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], "+", "\[Epsilon]"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List[RowBox[List["n", "!"]], " ", "\[Epsilon]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "q"], RowBox[List[FractionBox[RowBox[List[SubscriptBox["p", "k"], " ", SuperscriptBox["\[Epsilon]", "q"]]], RowBox[List[RowBox[List["(", RowBox[List["q", "-", "k"]], ")"]], "!"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "-", "1"]]]]], SuperscriptBox["j", "i"]]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "i"]]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["s", "1"], "\[Equal]", "EulerGamma"]], "\[And]", RowBox[List[SubscriptBox["s", "k"], "\[Equal]", RowBox[List["Zeta", "[", "k", "]"]]]]]]]], "/;", RowBox[List[RowBox[List["k", ">", "1"]], "\[And]", RowBox[List[SubscriptBox["a", "0"], "\[Equal]", RowBox[List["-", "EulerGamma"]]]], "\[And]", RowBox[List[SubscriptBox["a", "k"], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "1"]]], " "]], RowBox[List["k", "+", "1"]]], SubscriptBox["s", RowBox[List["k", "+", "1"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["k", ">", "0"]], "\[And]", RowBox[List[SubscriptBox["p", "0"], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["EulerGamma", "m"]]]]], "\[And]", RowBox[List[SubscriptBox["p", "k"], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SubscriptBox["a", "0"], " ", "k"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["m", " ", "j"]], "-", "k", "+", "j"]], ")"]], SubscriptBox["a", "j"], " ", SubscriptBox["p", RowBox[List["k", "-", "j"]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["k", ">", "0"]], "\[And]", "m"]]]], "=", RowBox[List["q", "-", "k"]]]]]]

 MathML Form

 Γ ( - n + ϵ ) ( - 1 ) n n ! ϵ q = 0 k = 0 q p k ϵ q ( q - k ) ! ( 1 + i = 1 ( j = 1 n ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - 1 ) j - 1 j i ) ϵ i ) /; n s 1 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] s k ζ ( k ) TagBox[RowBox[List["\[Zeta]", "(", TagBox["k", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] /; k > 1 a 0 - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] a k ( - 1 ) k + 1 s k + 1 k + 1 /; k > 0 p 0 ( - 1 ) m TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] m p k 1 a 0 k j = 1 k ( m j + j - k ) a j p k - j /; k > 0 m = q - k Set Condition Condition Condition Condition Proportional Gamma -1 n ϵ -1 n n ϵ -1 k 0 q q 0 Subscript p k ϵ q q -1 k -1 1 i 1 j 1 n Binomial n j -1 j -1 j i -1 ϵ i n Subscript s 1 Subscript s k Zeta k k 1 Subscript a 0 -1 Subscript a k -1 k 1 Subscript s k 1 k 1 -1 k 0 Subscript p 0 -1 m m Subscript p k 1 Subscript a 0 k -1 j 1 k m j j -1 k Subscript a j Subscript p k -1 j k 0 m q -1 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], "+", "\[Epsilon]"]], "]"]], "\[Proportional]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "q"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["p", "k"], " ", SuperscriptBox["\[Epsilon]", "q"]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "-", "1"]]]]], SuperscriptBox["j", "i"]]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "i"]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["q", "-", "k"]], ")"]], "!"]]]]]]]]], RowBox[List[RowBox[List["n", "!"]], " ", "\[Epsilon]"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["s", "1"], "\[Equal]", "EulerGamma"]], "&&", RowBox[List[SubscriptBox["s", "k"], "\[Equal]", RowBox[List["Zeta", "[", "k", "]"]]]]]]]], "/;", RowBox[List[RowBox[List["k", ">", "1"]], "&&", RowBox[List[SubscriptBox["a", "0"], "\[Equal]", RowBox[List["-", "EulerGamma"]]]], "&&", RowBox[List[SubscriptBox["a", "k"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "1"]]], " ", SubscriptBox["s", RowBox[List["k", "+", "1"]]]]], RowBox[List["k", "+", "1"]]]]]]]]], "/;", RowBox[List[RowBox[List["k", ">", "0"]], "&&", RowBox[List[SubscriptBox["p", "0"], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["EulerGamma", "m"]]]]], "&&", RowBox[List[SubscriptBox["p", "k"], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["m", " ", "j"]], "-", "k", "+", "j"]], ")"]], " ", SubscriptBox["a", "j"], " ", SubscriptBox["p", RowBox[List["k", "-", "j"]]]]]]], RowBox[List[SubscriptBox["a", "0"], " ", "k"]]]]]]]]], "/;", RowBox[List[RowBox[List["k", ">", "0"]], "&&", "m"]]]], "=", RowBox[List["q", "-", "k"]]]]]]

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

 2007-05-02