|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/06.08.10.0003.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
GammaRegularized[a, z] == (z^a/(E^z Gamma[a]))
(1/(1 - a + z + (a - 1)/(3 - a + z + (2 (a - 2))/
(5 - a + z + (3 (a - 3))/(7 - a + z + (4 (a - 4))/
(9 - a + z + (5 (a - 5))/(11 - a + z + \[Ellipsis]))))))) /;
!IntervalMemberQ[Interval[{-Infinity, 0}], z]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GammaRegularized", "[", RowBox[List["a", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]], RowBox[List["Gamma", "[", "a", "]"]]], FractionBox["1", RowBox[List["1", "-", "a", "+", "z", "+", FractionBox[RowBox[List["a", "-", "1"]], RowBox[List["3", "-", "a", "+", "z", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]]]], RowBox[List["5", "-", "a", "+", "z", "+", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List["a", "-", "3"]], ")"]]]], RowBox[List["7", "-", "a", "+", "z", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["a", "-", "4"]], ")"]]]], RowBox[List["9", "-", "a", "+", "z", "+", FractionBox[RowBox[List["5", " ", RowBox[List["(", RowBox[List["a", "-", "5"]], ")"]]]], RowBox[List["11", "-", "a", "+", "z", "+", "\[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> <semantics> <mi> Q </mi> <annotation-xml encoding='MathML-Content'> <ci> GammaRegularized </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mstyle scriptlevel='0'> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 5 </mn> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 7 </mn> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 9 </mn> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 11 </mn> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mo> … </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> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <semantics> <mi> Q </mi> <annotation-xml encoding='MathML-Content'> <ci> GammaRegularized </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mstyle scriptlevel='0'> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 5 </mn> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 7 </mn> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 9 </mn> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 11 </mn> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mo> … </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> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </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["GammaRegularized", "[", RowBox[List["a_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["(", RowBox[List["1", "-", "a", "+", "z", "+", FractionBox[RowBox[List["a", "-", "1"]], RowBox[List["3", "-", "a", "+", "z", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]]]], RowBox[List["5", "-", "a", "+", "z", "+", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List["a", "-", "3"]], ")"]]]], RowBox[List["7", "-", "a", "+", "z", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["a", "-", "4"]], ")"]]]], RowBox[List["9", "-", "a", "+", "z", "+", FractionBox[RowBox[List["5", " ", RowBox[List["(", RowBox[List["a", "-", "5"]], ")"]]]], RowBox[List["11", "-", "a", "+", "z", "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]], ")"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|