Incomplete beta function: Series representations (formula 06.19.06.0062)

Beta

$Mathematica Notation$

Gamma, Beta, Erf

Beta[z,a,b]

Series representations

Generalized power series

Expansions at z==infinity

http://functions.wolfram.com/06.19.06.0062.01

Input Form

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

Standard Form

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MathML Form

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Rule Form

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

2007-05-02

Beta[a,b]
Beta[z₁,z₂,a,b]