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variants of this functions
BetaRegularized




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > BetaRegularized[z,a,b] > Integral representations > Contour integral representations





http://functions.wolfram.com/06.21.07.0004.01









  


  








Input Form





BetaRegularized[z, a, b] == (1/(2 Pi I)) ((Sin[Pi b] z^a)/(Pi Gamma[a])) ContourIntegrate[(Gamma[s] Gamma[s + b] Gamma[a - s] Gamma[1 - b - s])/ (1 - z)^s, {s, \[ScriptCapitalL]}] /; Abs[Arg[1 - z]] < Pi








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BetaRegularized", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "b"]], "]"]], " ", SuperscriptBox["z", "a"]]], ")"]], " ", RowBox[List["ContourIntegrate", "[", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "b", "-", "s"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["-", "s"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "a", "]"]]]], ")"]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["1", "-", "z"]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]


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





2001-10-29