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http://functions.wolfram.com/06.08.10.0004.01
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GammaRegularized[a, z] == z^a/(E^z Gamma[a])/
(z - a + 1 + ContinueFraction[{(-k) (k - a), z + 1 - a + 2 k},
{k, 1, Infinity}]) /; !IntervalMemberQ[Interval[{-Infinity, 0}], z]
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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", "]"]]], "/", RowBox[List["(", RowBox[List["z", "-", "a", "+", "1", "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "k"]], " ", RowBox[List["(", RowBox[List["k", "-", "a"]], ")"]]]], ",", RowBox[List["z", "+", "1", "-", "a", "+", RowBox[List["2", " ", "k"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]]]], ")"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]
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<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> <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> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> <mo> + </mo> <msubsup> <mrow> <msub> <mi> Κ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 1 </mn> <mi> ∞ </mi> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </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> <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> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> <mo> + </mo> <msubsup> <mrow> <msub> <mi> Κ </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 1 </mn> <mi> ∞ </mi> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </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>
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| 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["z", "-", "a", "+", "1", "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "k"]], " ", RowBox[List["(", RowBox[List["k", "-", "a"]], ")"]]]], ",", RowBox[List["z", "+", "1", "-", "a", "+", RowBox[List["2", " ", "k"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]]]], ")"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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