Incomplete beta function: Series representations (formula 06.19.06.0064)

Beta

$Mathematica Notation$

Gamma, Beta, Erf

Beta[z,a,b]

Series representations

Generalized power series

Expansions at z==infinity

http://functions.wolfram.com/06.19.06.0064.01

Input Form

Beta[z, a, b] \[Proportional] Piecewise[{{ComplexInfinity, a == 0 || (Element[-a, Integers] && -a > 0 && Element[b, Integers] && b > 0 && a + b > 0)}, {Log[z]/E^(I a Pi), Arg[z] <= 0 && a + b == 1}, {(Gamma[a] Gamma[1 - a - b])/(E^(I a Pi) Gamma[1 - b]), Arg[z] <= 0 && Re[a + b] < 1}, {(E^(I b Pi) z^(a + b - 1))/(1 - a - b), (Arg[z] <= 0 && Re[a + b] > 1) || (Element[-a, Integers] && -a > 0 && Element[b, Integers] && b > 0 && a + b <= 0)}, {(E^(I b Pi) z^(a + b - 1))/(1 - a - b) + (Gamma[a] Gamma[1 - a - b])/ (E^(I a Pi) Gamma[1 - b]), Arg[z] <= 0}, {E^(I a Pi) Log[z], Arg[z] > 0 && a + b == 1}, {(E^(I a Pi) Gamma[a] Gamma[1 - a - b])/ Gamma[1 - b], Arg[z] > 0 && Re[a + b] < 1}, {z^(a + b - 1)/(E^(I b Pi) (1 - a - b)), (Arg[z] > 0 && Re[a + b] > 1) || (Element[-a, Integers] && -a > 0 && Element[b, Integers] && b > 0 && a + b <= 0)}}, z^(a + b - 1)/(E^(I b Pi) (1 - a - b)) + (E^(I a Pi) Gamma[a] Gamma[1 - a - b])/Gamma[1 - b]] /; (Abs[z] -> Infinity)

Standard Form

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MathML Form

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

2007-05-02

Beta[a,b]
Beta[z₁,z₂,a,b]