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http://functions.wolfram.com/06.08.06.0032.01
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GammaRegularized[a, z] \[Proportional]
1 - ((a^(-(1/2) - a) E^(a - z) z^a)/Sqrt[2 Pi]) (1 + (-1 + 12 z)/(12 a) +
(1 - 312 z + 288 z^2)/(288 a^2) + O[1/a^3]) /; (Abs[a] -> Infinity)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GammaRegularized", "[", RowBox[List["a", ",", "z"]], "]"]], "\[Proportional]", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "a"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["a", "-", "z"]]], " ", SuperscriptBox["z", "a"]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["12", " ", "z"]]]], RowBox[List["12", " ", "a"]]], "+", FractionBox[RowBox[List["1", "-", RowBox[List["312", " ", "z"]], "+", RowBox[List["288", " ", SuperscriptBox["z", "2"]]]]], RowBox[List["288", " ", SuperscriptBox["a", "2"]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["a", "3"]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "a", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]
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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> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mi> a </mi> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> a </mi> <mo> - </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> a </mi> </msup> </mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mn> 288 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 312 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 288 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msup> <mi> a </mi> <mn> 3 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> a </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> GammaRegularized </ci> <ci> a </ci> <ci> z </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 288 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 312 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 288 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> a </ci> </apply> <infinity /> </apply> </apply> </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[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "a"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["a", "-", "z"]]], " ", SuperscriptBox["z", "a"]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["12", " ", "z"]]]], RowBox[List["12", " ", "a"]]], "+", FractionBox[RowBox[List["1", "-", RowBox[List["312", " ", "z"]], "+", RowBox[List["288", " ", SuperscriptBox["z", "2"]]]]], RowBox[List["288", " ", SuperscriptBox["a", "2"]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["a", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "3"]], "]"]]]], ")"]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "a", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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