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 BetaRegularized

 http://functions.wolfram.com/06.21.06.0055.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["b", "-", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "-", "1"]], ")"]], RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]], " "]]], "-", RowBox[List[FractionBox[RowBox[List["Sin", "[", RowBox[List["a", " ", "\[Pi]"]], "]"]], "\[Pi]"], SuperscriptBox["z", "a"], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "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"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["a", "+", "b", "-", "1"]], ">", "0"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[FractionBox[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "a"]], "]"]], "\[Pi]"], SuperscriptBox["z", "a"], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List["PolyGamma", "[", "a", "]"]], "-", "EulerGamma"]], ")"]]]], "-", FractionBox[RowBox[List["a", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "a"]], "]"]], SuperscriptBox["z", RowBox[List["a", "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]]]], "\[Pi]"]]], ",", RowBox[List[RowBox[List["a", "+", "b"]], "\[Equal]", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["-", "a"]], ")"]], "!"]], " "]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "b"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "!"]]]]]]], FractionBox[SuperscriptBox["z", RowBox[List["a", "+", "b", "-", "1"]]], RowBox[List["(", 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"]]]]]], "}"]], ",", "\[IndentingNewLine]", RowBox[List["{", RowBox[List["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"]], ">", "0"]]]]]], "}"]]]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], SuperscriptBox["z", "a"], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", "a"], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["b", "-", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "-", "1"]], ")"]], RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 I BetaRegularized z ( a , b ) z a ( - z ) b - 1 ( a + b - 1 ) Β Beta ( a , b ) - sin ( a π ) z a ( - z ) - a ( log ( - z ) - ψ TagBox["\[Psi]", PolyGamma] ( a ) + ψ TagBox["\[Psi]", PolyGamma] ( a + b ) ) π a + b - 1 + sin ( π a ) z a ( - z ) - a ( log ( - z ) - ψ TagBox["\[Psi]", PolyGamma] ( a ) - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) π - a sin ( π a ) z a - 1 ( - z ) - a π a + b 1 z a ( - z ) b - 1 ( a + b - 1 ) Β Beta ( a , b ) + csc ( π ( a + b ) ) sin ( b π ) z a ( - z ) - a True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional BetaRegularized z a b z a -1 z b -1 a b -1 Beta a b -1 -1 a z a -1 z -1 a -1 z -1 PolyGamma a PolyGamma a b -1 a b -1 SuperPlus a z a -1 z -1 a -1 z -1 PolyGamma a -1 -1 -1 a a z a -1 -1 z -1 a -1 a b 1 z a -1 z b -1 a b -1 Beta a b -1 a b b z a -1 z -1 a Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BetaRegularized", "[", 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[RowBox[List["(", RowBox[List["a", "+", "b", "-", "1"]], ")"]], " ", RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["a", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox["z", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List["PolyGamma", "[", "a", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "b"]], "]"]]]], ")"]]]], "\[Pi]"]]], RowBox[List[RowBox[List[RowBox[List["a", "+", "b", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", "+", "b", "-", "1"]], ">", "0"]]]]], List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "a"]], "]"]], " ", SuperscriptBox["z", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List["PolyGamma", "[", "a", "]"]], "-", "EulerGamma"]], ")"]]]], "\[Pi]"], "-", FractionBox[RowBox[List["a", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "a"]], "]"]], " ", SuperscriptBox["z", RowBox[List["a", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]]]], "\[Pi]"]]], RowBox[List[RowBox[List["a", "+", "b"]], "\[Equal]", "1"]]], List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["-", "a"]], ")"]], "!"]], " ", SuperscriptBox["z", RowBox[List["a", "+", "b", "-", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "b"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], "!"]]]], ")"]], " ", RowBox[List["(", 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["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"]], ">", "0"]]]]], List[RowBox[List[RowBox[List[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox["z", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["b", "-", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "-", "1"]], ")"]], " ", RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]]]]]], 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