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

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2007-05-02