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variants of this functions
Beta




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Beta[z,a,b] > Series representations > Generalized power series > Expansions at z==infinity





http://functions.wolfram.com/06.19.06.0061.01









  


  








Input Form





Beta[z, a, b] \[Proportional] Piecewise[{{(z^a (-z)^(b - 1))/(a + b - 1) + ((Pochhammer[1 - a, a + b - 1]/(a + b - 1)!) z^a (Log[-z] - PolyGamma[a] + PolyGamma[a + b]))/(-z)^a, Element[a + b - 1, Integers] && a + b - 1 > 0}, {(z^a (Log[-z] - PolyGamma[a] - EulerGamma))/(-z)^a, a + b == 1}, {((-1)^(b - 1)/(a + b - 1)) z^(a + b - 1), Element[-a, Integers] && -a > 0 && Element[b, Integers] && b > 0 && a + b <= 0}, {ComplexInfinity, a == 0 || (Element[-a, Integers] && -a > 0 && Element[b, Integers] && b > 0 && a + b > 0)}}, (Gamma[a] Gamma[1 - a - b] z^a)/((-z)^a Gamma[1 - b]) + (z^a (-z)^(-1 + b))/(a + b - 1)] /; (Abs[z] -> Infinity)








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Beta", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["b", "-", "1"]]]]], RowBox[List["a", "+", "b", "-", "1"]]], "+", FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "a"]], ",", RowBox[List["a", "+", "b", "-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", SuperscriptBox["z", "a"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List["PolyGamma", "[", "a", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "b"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "-", "1"]], ")"]], "!"]]]]], RowBox[List[RowBox[List[RowBox[List["a", "+", "b", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "-", "1"]], ">", "0"]]]]], List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", SuperscriptBox["z", "a"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List["PolyGamma", "[", "a", "]"]], "-", "EulerGamma"]], ")"]]]], RowBox[List[RowBox[List["a", "+", "b"]], "\[Equal]", "1"]]], List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["b", "-", "1"]]], " ", SuperscriptBox["z", RowBox[List["a", "+", "b", "-", "1"]]]]], RowBox[List["a", "+", "b", "-", "1"]]], RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "a"]], ">", "0"]], "&&", RowBox[List["b", "\[Element]", "Integers"]], "&&", RowBox[List["b", ">", "0"]], "&&", RowBox[List[RowBox[List["a", "+", "b"]], "\[LessEqual]", "0"]]]]], List["ComplexInfinity", RowBox[List[RowBox[List["a", "\[Equal]", "0"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "a"]], ">", "0"]], "&&", RowBox[List["b", "\[Element]", "Integers"]], "&&", RowBox[List["b", ">", "0"]], "&&", RowBox[List[RowBox[List["a", "+", "b"]], ">", "0"]]]], ")"]]]]], List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "-", "b"]], "]"]], " ", SuperscriptBox["z", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "b"]]]]], RowBox[List["a", "+", "b", "-", "1"]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]


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





2007-05-02