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variants of this functions
Gamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Gamma[a,z1,z2] > Series representations > Generalized power series > Expansions at {z1,z2}=={0,0} > For the function itself > Special cases





http://functions.wolfram.com/06.07.06.0013.01









  


  








Input Form





Gamma[-n, Subscript[z, 1], Subscript[z, 2]] == ((-1)^(n - 1)/n!) (Log[Subscript[z, 1]] - Log[Subscript[z, 2]]) - Sum[If[k == n, 0, ((-1)^k (Subscript[z, 1]^(k - n) - Subscript[z, 2]^(k - n)))/((k - n) k!)], {k, 0, Infinity}] /; Element[n, Integers] && n >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " "]], RowBox[List["n", "!"]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]], "-", RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]]]], ")"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List["If", "[", RowBox[List[RowBox[List["k", "\[Equal]", "n"]], ",", "0", ",", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["(", RowBox[List[SubsuperscriptBox["z", "1", RowBox[List["k", "-", "n"]]], "-", SubsuperscriptBox["z", "2", RowBox[List["k", "-", "n"]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "n"]], ")"]], " ", RowBox[List["k", "!"]]]]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <munder> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> k </mi> <mo> &#8800; </mo> <mi> n </mi> </mrow> </munder> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 1 </mn> <mrow> <mi> k </mi> <mo> - </mo> <mi> n </mi> </mrow> </msubsup> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mrow> <mi> k </mi> <mo> - </mo> <mi> n </mi> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n_"]], ",", SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]], "-", RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]]]], ")"]]]], RowBox[List["n", "!"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List["If", "[", RowBox[List[RowBox[List["k", "\[Equal]", "n"]], ",", "0", ",", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[SubsuperscriptBox["zz", "1", RowBox[List["k", "-", "n"]]], "-", SubsuperscriptBox["zz", "2", RowBox[List["k", "-", "n"]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "n"]], ")"]], " ", RowBox[List["k", "!"]]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2007-05-02