html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Beta

 http://functions.wolfram.com/06.19.06.0025.01

 Input Form

 Beta[z, a, b] \[Proportional] (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)])) (MeijerG[{{1 - a, b}, {}}, {{0, b}, {}}, 1 - Subscript[z, 0]] (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)]) - (2 I E^(I b Pi Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)]) Pi Floor[(Pi + Arg[1 - Subscript[z, 0]])/(2 Pi)] Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)] Gamma[a])/Subscript[z, 0]^a + Subscript[z, 0]^(-1 - a) (-2 I E^(I b Pi Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)]) Pi Floor[(Pi + Arg[1 - Subscript[z, 0]])/(2 Pi)] Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)] (a Gamma[a] - Gamma[1 + a]) + (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)]) Subscript[z, 0]^a (a MeijerG[{{1 - a, b}, {}}, {{0, b}, {}}, 1 - Subscript[z, 0]] + MeijerG[{{-a, -1 + b}, {}}, {{0, -1 + b}, {}}, 1 - Subscript[z, 0]] Subscript[z, 0])) (z - Subscript[z, 0]) + (1/2) Subscript[z, 0]^(-2 - a) (-2 I E^(I b Pi Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)]) Pi Floor[(Pi + Arg[1 - Subscript[z, 0]])/(2 Pi)] Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)] ((-1 + a) a Gamma[a] - 2 a Gamma[1 + a] + Gamma[2 + a]) + (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)]) Subscript[z, 0]^a ((-1 + a) a MeijerG[{{1 - a, b}, {}}, {{0, b}, {}}, 1 - Subscript[z, 0]] + Subscript[z, 0] (2 a MeijerG[{{-a, -1 + b}, {}}, {{0, -1 + b}, {}}, 1 - Subscript[z, 0]] + MeijerG[{{-1 - a, -2 + b}, {}}, {{0, -2 + b}, {}}, 1 - Subscript[z, 0]] Subscript[z, 0]))) (z - Subscript[z, 0])^2 + O[(z - Subscript[z, 0])^3])

 Standard Form

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 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 ) ( - 2 b π arg ( z 0 - z ) 2 π π arg ( 1 - z 0 ) + π 2 π arg ( z 0 - z ) 2 π Γ ( a ) z 0 - a + G 2 , 2 2 , 2 ( 1 - z 0 1 - a , b 0 , b ) 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["1", "-", "a"]], MeijerG, Rule[Editable, True]], ",", TagBox["b", MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox["b", MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ( 1 1 - z 0 ) b arg ( z 0 - z ) 2 π ( 1 - z 0 ) b arg ( z 0 - z ) 2 π + ( ( 1 1 - z 0 ) b arg ( z 0 - z ) 2 π ( 1 - z 0 ) b arg ( z 0 - z ) 2 π z 0 a ( a G 2 , 2 2 , 2 ( 1 - z 0 1 - a , b 0 , b ) 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["1", "-", "a"]], MeijerG, Rule[Editable, True]], ",", TagBox["b", MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox["b", MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + G 2 , 2 2 , 2 ( 1 - z 0 - a , b - 1 0 , b - 1 ) 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["-", "a"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "1"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] z 0 ) - 2 b π arg ( z 0 - z ) 2 π π arg ( 1 - z 0 ) + π 2 π arg ( z 0 - z ) 2 π ( a Γ ( a ) - Γ ( a + 1 ) ) ) z 0 - a - 1 ( z - z 0 ) + 1 2 ( ( 1 1 - z 0 ) b arg ( z 0 - z ) 2 π ( 1 - z 0 ) b arg ( z 0 - z ) 2 π z 0 a ( ( a - 1 ) a G 2 , 2 2 , 2 ( 1 - z 0 1 - a , b 0 , b ) 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], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", "a"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["b", MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["b", MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] + z 0 ( 2 a G 2 , 2 2 , 2 ( 1 - z 0 - a , b - 1 0 , b - 1 ) 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], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["-", "a"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["b", "-", "1"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["b", "-", "1"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] + G 2 , 2 2 , 2 ( 1 - z 0 - a - 1 , b - 2 0 , b - 2 ) 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], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "1"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["b", "-", "2"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["b", "-", "2"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] z 0 ) ) - 2 b π arg ( z 0 - z ) 2 π π arg ( 1 - z 0 ) + π 2 π arg ( z 0 - z ) 2 π ( ( a - 1 ) a Γ ( a ) - 2 a Γ ( a + 1 ) + Γ ( a + 2 ) ) ) z 0 - a - 2 ( z - z 0 ) 2 + O ( ( z - z 0 ) 3 ) ) Proportional 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 -2 b Subscript z 0 -1 z 2 -1 1 -1 Subscript z 0 2 -1 Subscript z 0 -1 z 2 -1 Gamma a Subscript z 0 -1 a MeijerG 1 -1 a b 0 b 1 -1 Subscript z 0 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 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 Subscript z 0 a a MeijerG 1 -1 a b 0 b 1 -1 Subscript z 0 MeijerG -1 a b -1 0 b -1 1 -1 Subscript z 0 Subscript z 0 -1 2 b Subscript z 0 -1 z 2 -1 1 -1 Subscript z 0 2 -1 Subscript z 0 -1 z 2 -1 a Gamma a -1 Gamma a 1 Subscript z 0 -1 a -1 z -1 Subscript z 0 1 2 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 Subscript z 0 a a -1 a MeijerG 1 -1 a b 0 b 1 -1 Subscript z 0 Subscript z 0 2 a MeijerG -1 a b -1 0 b -1 1 -1 Subscript z 0 MeijerG -1 a -1 b -2 0 b -2 1 -1 Subscript z 0 Subscript z 0 -1 2 b Subscript z 0 -1 z 2 -1 1 -1 Subscript z 0 2 -1 Subscript z 0 -1 z 2 -1 a -1 a Gamma a -1 2 a Gamma a 1 Gamma a 2 Subscript z 0 -1 a -2 z -1 Subscript z 0 2 O z -1 Subscript z 0 3 [/itex]

 Rule Form

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

 2007-05-02