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 Beta

 http://functions.wolfram.com/06.19.06.0036.01

 Input Form

 Beta[z, a, b] == (1/(Gamma[1 - b] Gamma[a + b])) Sum[(((-1)^(k - j) Pochhammer[-a, k - j] x^(j - k))/(j! (k - j)!)) ((-2 Pi I Floor[Arg[x - z]/(2 Pi)] E^(I b Pi Floor[Arg[x - z]/(2 Pi)]) (-1)^j Gamma[a + j])/x^j + x^a (1/(1 - x))^(b Floor[Arg[-z + x]/(2 Pi)]) (1 - x)^(b Floor[Arg[-z + x]/(2 Pi)]) MeijerG[{{1 - a - j, b - j}, {}}, {{0, b - j}, {}}, 1 - x]) (z - x)^k, {k, 0, Infinity}, {j, 0, k}] /; Element[x, Reals] && x > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Beta", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b"]], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[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[RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "b", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "j"]], "]"]], " ", SuperscriptBox["x", RowBox[List["-", "j"]]]]], "+", RowBox[List[SuperscriptBox["x", "a"], SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", "x"]]], ")"]], RowBox[List["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "x"]], ")"]], RowBox[List["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], 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["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", ">", "1"]]]]]]]]

 MathML Form

 Β Beta z ( a , b ) 1 Γ ( 1 - b ) Γ ( a + b ) k = 0 j = 0 k ( - 1 ) k - j ( - a ) k - j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "a"]], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] x j - k j ! ( k - j ) ! ( x a ( 1 1 - x ) b arg ( x - z ) 2 π ( 1 - x ) b arg ( x - z ) 2 π G 2 , 2 2 , 2 ( 1 - x - a - j + 1 , b - j 0 , b - j ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["1", "-", "x"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "j", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "j"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "j"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] - 2 π arg ( x - z ) 2 π b π arg ( x - z ) 2 π ( - 1 ) j Γ ( a + j ) x - j ) ( z - x ) k /; x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x > 1 Condition Beta z a b 1 Gamma 1 -1 b Gamma a b -1 j 0 k k 0 -1 k -1 j Pochhammer -1 a k -1 j x j -1 k j k -1 j -1 x a 1 1 -1 x -1 b x -1 z 2 -1 1 -1 x b x -1 z 2 -1 MeijerG -1 a -1 j 1 b -1 j 0 b -1 j 1 -1 x -1 2 x -1 z 2 -1 b x -1 z 2 -1 -1 j Gamma a j x -1 j z -1 x k x x 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Beta", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[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["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "b", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "j"]], "]"]], " ", SuperscriptBox["x", RowBox[List["-", "j"]]]]], "+", RowBox[List[SuperscriptBox["x", "a"], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", "x"]]], ")"]], RowBox[List["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "x"]], ")"]], RowBox[List["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", 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[RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b"]], "]"]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", ">", "1"]]]]]]]]]]

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

 2007-05-02