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variants of this functions
Gamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Gamma[a,z] > Series representations > Generalized power series > Expansions at generic point a==a0 > For the function itself





http://functions.wolfram.com/06.06.06.0019.01









  


  








Input Form





Gamma[a, z] == Sum[(Derivative[k][Gamma][Subscript[a, 0]]/k! - z^Subscript[a, 0] Sum[((-1)^(k - j)/j!) Gamma[Subscript[a, 0]]^ (k - j + 1) Log[z]^j HypergeometricPFQRegularized[ {Subscript[c, 1], Subscript[c, 2], \[Ellipsis], Subscript[c, k - j + 1]}, {1 + Subscript[c, 1], 1 + Subscript[c, 2], \[Ellipsis], 1 + Subscript[c, k - j + 1]}, -z], {j, 0, k}]) (a - Subscript[a, 0])^k, {k, 0, Infinity}] /; Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, k + 1] == Subscript[a, 0] && Element[k, Integers] && k >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", SubscriptBox["a", "0"], "]"]], RowBox[List["k", "!"]]], "-", RowBox[List[SuperscriptBox["z", SubscriptBox["a", "0"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], " "]], RowBox[List["j", "!"]]], SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["a", "0"], "]"]], RowBox[List["k", "-", "j", "+", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "j"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["c", "1"], ",", SubscriptBox["c", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["c", RowBox[List["k", "-", "j", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["c", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["c", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["c", RowBox[List["k", "-", "j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["a", "0"]]], ")"]], "k"]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "1"], "\[Equal]", SubscriptBox["c", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["c", RowBox[List["k", "+", "1"]]], "\[Equal]", SubscriptBox["a", "0"]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", "\[GreaterEqual]", "0"]]]]]]]]








MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Gamma", "[", RowBox[List["a_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", SubscriptBox["aa", "0"], "]"]], RowBox[List["k", "!"]]], "-", RowBox[List[SuperscriptBox["z", SubscriptBox["aa", "0"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], RowBox[List["k", "-", "j", "+", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "j"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["c", "1"], ",", SubscriptBox["c", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["c", RowBox[List["k", "-", "j", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["c", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["c", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["c", RowBox[List["k", "-", "j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]], RowBox[List["j", "!"]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], ")"]], "k"]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "1"], "\[Equal]", SubscriptBox["c", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["c", RowBox[List["k", "+", "1"]]], "\[Equal]", SubscriptBox["aa", "0"]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2007-05-02