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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/06.06.06.0017.01









  


  








Input Form





Gamma[a, z] \[Proportional] Gamma[Subscript[a, 0], z] + (Gamma[Subscript[a, 0]] (-Log[z] + PolyGamma[Subscript[a, 0]]) + (z^Subscript[a, 0]/Subscript[a, 0]^2) HypergeometricPFQ[ {Subscript[a, 0], Subscript[a, 0]}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, -z] + Log[z] Gamma[Subscript[a, 0], z]) (a - Subscript[a, 0]) - (1/2) ((-Gamma[Subscript[a, 0], z]) Log[z]^2 + Gamma[Subscript[a, 0]] (Log[z]^2 - PolyGamma[Subscript[a, 0]]^2 - PolyGamma[1, Subscript[a, 0]]) + ((2 z^Subscript[a, 0])/Subscript[a, 0]^3) (HypergeometricPFQ[{Subscript[a, 0], Subscript[a, 0], Subscript[a, 0]}, {1 + Subscript[a, 0], 1 + Subscript[a, 0], 1 + Subscript[a, 0]}, -z] - Subscript[a, 0] Log[z] HypergeometricPFQ[{Subscript[a, 0], Subscript[a, 0]}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, -z])) (a - Subscript[a, 0])^2 + \[Ellipsis] /; (a -> Subscript[a, 0])








Standard Form





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MathML Form







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<apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> z </ci> </apply> <apply> <power /> <apply> <ln /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ln /> <ci> z </ci> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Gamma", "[", RowBox[List["a_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["aa", "0"], ",", "z"]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "0"], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", SubscriptBox["aa", "0"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]], SubsuperscriptBox["aa", "0", "2"]], "+", RowBox[List[RowBox[List["Log", "[", "z", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["aa", "0"], ",", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], ")"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["aa", "0"], ",", "z"]], "]"]]]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "2"]]], "+", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "2"], "-", SuperscriptBox[RowBox[List["PolyGamma", "[", SubscriptBox["aa", "0"], "]"]], "2"], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", SubscriptBox["aa", "0"]]], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["z", SubscriptBox["aa", "0"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]], "-", RowBox[List[SubscriptBox["aa", "0"], " ", RowBox[List["Log", "[", "z", "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]], ")"]]]], SubsuperscriptBox["aa", "0", "3"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["a", "\[Rule]", SubscriptBox["aa", "0"]]], ")"]]]]]]]]


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





2007-05-02