Gamma[a, z] == Gamma[a] - z^a/(E^z (a - (a z)/(a + 1 + z/(a + 2 - ((a + 1) z)/ (a + 3 + (2 z)/(a + 4 - ((a + 2) z)/(a + 5 + \[Ellipsis]))))))) /; !IntervalMemberQ[Interval[{-Infinity, 0}], z]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], "-", FractionBox[RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]], RowBox[List["a", "-", FractionBox[RowBox[List["a", " ", "z"]], RowBox[List["a", "+", "1", "+", FractionBox["z", RowBox[List["a", "+", "2", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "1"]], ")"]], " ", "z"]], RowBox[List["a", "+", "3", "+", FractionBox[RowBox[List["2", "z"]], RowBox[List["a", "+", "4", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "2"]], ")"]], " ", "z"]], RowBox[List["a", "+", "5", "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]

<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> <mi> a </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mstyle scriptlevel='0'> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> <mo> + </mo> <mfrac> <mi> z </mi> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mn> 3 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mn> 4 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mn> 5 </mn> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mstyle> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mstyle scriptlevel='0'> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> <mo> + </mo> <mfrac> <mi> z </mi> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mn> 3 </mn> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mn> 4 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mn> 5 </mn> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mstyle> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Gamma", "[", RowBox[List["a_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], "-", FractionBox[RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]], RowBox[List["a", "-", FractionBox[RowBox[List["a", " ", "z"]], RowBox[List["a", "+", "1", "+", FractionBox["z", RowBox[List["a", "+", "2", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "1"]], ")"]], " ", "z"]], RowBox[List["a", "+", "3", "+", FractionBox[RowBox[List["2", " ", "z"]], RowBox[List["a", "+", "4", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "2"]], ")"]], " ", "z"]], RowBox[List["a", "+", "5", "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]

2001-10-29