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 Beta

 http://functions.wolfram.com/06.19.06.0026.01

 Input Form

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

 Standard Form

 Cell[BoxData[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"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["z", "0"]], ")"]], RowBox[List["a", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["z", "0", RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]], " ", 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"]]]], "]"]], " ", SubsuperscriptBox["z", "0", 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[SubscriptBox["z", "0"], "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "b", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[SubscriptBox["z", "0"], "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "-", SubscriptBox["z", "0"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "j"]], "]"]], " ", SubsuperscriptBox["z", "0", RowBox[List[RowBox[List["-", "a"]], "-", "j"]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", SubscriptBox["z", "0"]]]], ")"]], RowBox[List["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["z", "0"]]], ")"]], RowBox[List["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["z", "0"]]], "]"]], 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", "-", SubscriptBox["z", "0"]]]]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]]]

 MathML Form

 Β Beta z ( a , b ) 1 Γ ( 1 - b ) Γ ( a + b ) ( 1 z 0 ) a arg ( z - z 0 ) 2 π z 0 a ( arg ( z - z 0 ) 2 π + 1 ) k = 0 j = 0 k ( - 1 ) k - j ( - a ) k - j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "a"]], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] z 0 j - k j ! ( k - j ) ! ( ( 1 1 - z 0 ) b arg ( z 0 - z ) 2 π ( 1 - z 0 ) b arg ( z 0 - z ) 2 π G 2 , 2 2 , 2 ( 1 - z 0 - 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", "-", SubscriptBox["z", "0"]]], 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 ( z 0 - z ) 2 π b π arg ( z 0 - z ) 2 π arg ( 1 - z 0 ) + π 2 π ( - 1 ) j Γ ( a + j ) z 0 - a - j ) ( z - z 0 ) k Beta z a b 1 Gamma 1 -1 b Gamma a b -1 1 Subscript z 0 -1 a z -1 Subscript z 0 2 -1 Subscript z 0 a z -1 Subscript z 0 2 -1 1 j 0 k k 0 -1 k -1 j Pochhammer -1 a k -1 j Subscript z 0 j -1 k j k -1 j -1 1 1 -1 Subscript z 0 -1 b Subscript z 0 -1 z 2 -1 1 -1 Subscript z 0 b Subscript z 0 -1 z 2 -1 MeijerG -1 a -1 j 1 b -1 j 0 b -1 j 1 -1 Subscript z 0 -1 2 Subscript z 0 -1 z 2 -1 b Subscript z 0 -1 z 2 -1 1 -1 Subscript z 0 2 -1 -1 j Gamma a j Subscript z 0 -1 a -1 j z -1 Subscript z 0 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Beta", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["zz", "0"]], ")"]], RowBox[List["a", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["zz", "0", RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], 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"]]]], "]"]], " ", SubsuperscriptBox["zz", "0", RowBox[List["j", "-", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[SubscriptBox["zz", "0"], "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "b", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[SubscriptBox["zz", "0"], "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "-", SubscriptBox["zz", "0"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "j"]], "]"]], " ", SubsuperscriptBox["zz", "0", RowBox[List[RowBox[List["-", "a"]], "-", "j"]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", SubscriptBox["zz", "0"]]]], ")"]], RowBox[List["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["zz", "0"]]], ")"]], RowBox[List["b", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", SubscriptBox["zz", "0"]]], "]"]], 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", "-", SubscriptBox["zz", "0"]]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "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"]], "]"]]]]]]]]]

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

 2007-05-02