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variants of this functions
GammaRegularized






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > GammaRegularized[a,z] > Continued fraction representations





http://functions.wolfram.com/06.08.10.0002.01









  


  










Input Form





GammaRegularized[a, z] == z^a/(E^z Gamma[a])/ (z + ContinueFraction[{2^((-1 - (-1)^k)/2) k^((1 + (-1)^k)/2) ((1 + k)/2 - a)^((1 - (-1)^k)/2), z^(((-1)^k + 1)/2)}, {k, 1, Infinity}]) /; !IntervalMemberQ[Interval[{-Infinity, 0}], z]










Standard Form





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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> <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> &#10869; </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> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <msubsup> <mrow> <msub> <mi> &#922; </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; 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Rule Form





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", "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], ")"]]]]], " ", SuperscriptBox["k", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["1", "+", "k"]], "2"], "-", "a"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], ")"]]]]]]], ",", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], "+", "1"]], ")"]]]]]]], "}"]], ",", 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"]], "]"]]]]]]]]]]










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





2001-10-29





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