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

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