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variants of this functions
BetaRegularized




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > BetaRegularized[z,a,b] > Series representations > Generalized power series > Expansions on branch cuts > For the function itself > In the left half-plane





http://functions.wolfram.com/06.21.06.0028.01









  


  








Input Form





BetaRegularized[z, a, b] == (Sin[Pi b]/(Pi Gamma[a])) (1/x)^(a Floor[Arg[z - x]/(2 Pi)]) x^(a (1 + Floor[Arg[z - x]/(2 Pi)])) Sum[(((-1)^(k - j) Pochhammer[-a, k - j] x^(j - k))/(j! (k - j)!)) MeijerG[{{1 - a - j, b - j}, {}}, {{0, b - j}, {}}, 1 - x] (z - x)^k, {k, 0, Infinity}, {j, 0, k}] /; Element[x, Reals] && x < 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_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "b"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", "x"], ")"]], RowBox[List["a", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox["x", RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "a"]], ",", RowBox[List["k", "-", "j"]]]], "]"]], " ", SuperscriptBox["x", RowBox[List["j", "-", "k"]]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "a", "-", "j"]], ",", RowBox[List["b", "-", "j"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["b", "-", "j"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["1", "-", "x"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]]]]]]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "a", "]"]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]]


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





2007-05-02