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variants of this functions
BetaRegularized






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > BetaRegularized[z,a,b] > Series representations > Residue representations





http://functions.wolfram.com/06.21.06.0065.01









  


  










Input Form





BetaRegularized[z, a, 1 - a + n] == (((-1)^(n - 1) n! Sin[a Pi])/(Pi Gamma[a])) z^a (Sum[Residue[(Gamma[s]/((-z)^s (a - s))) Gamma[a - n - s], {s, a - n + j}], {j, 0, n - 1}] + Sum[Residue[(Gamma[s]/((-z)^s (a - s))) Gamma[a - n - s], {s, a - n + j}], {j, n + 1, Infinity}] + Residue[(Gamma[s] (Gamma[a - n - s]/(a - s)))/(-z)^s, {s, a}]) /; Abs[z] > 1 && Element[n, Integers] && n >= 0










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_", ",", RowBox[List["1", "-", "a_", "+", "n_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["n", "!"]], " ", RowBox[List["Sin", "[", RowBox[List["a", " ", "\[Pi]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "a"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "-", "n", "-", "s"]], "]"]]]], RowBox[List["a", "-", "s"]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["a", "-", "n", "+", "j"]]]], "}"]]]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", RowBox[List["n", "+", "1"]]]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "-", "n", "-", "s"]], "]"]]]], RowBox[List["a", "-", "s"]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["a", "-", "n", "+", "j"]]]], "}"]]]], "]"]]]], "+", RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "s"]]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "-", "n", "-", "s"]], "]"]]]], RowBox[List["a", "-", "s"]]], ",", RowBox[List["{", RowBox[List["s", ",", "a"]], "}"]]]], "]"]]]], ")"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "a", "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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