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 BetaRegularized

 http://functions.wolfram.com/06.21.06.0056.01

 Input Form

 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)

 Standard Form

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 MathML Form

 I BetaRegularized z ( a , b ) - a π ( ( a + b ) π z a + b - 1 ( 1 - a - b ) Β Beta ( a , b ) - sin ( a π ) ( π + log ( z ) - ψ TagBox["\[Psi]", PolyGamma] ( a ) + ψ TagBox["\[Psi]", PolyGamma] ( a + b ) ) π ) arg ( z ) 0 a + b - 1 + - - a π sin ( a π ) ( a - π z + z ( - log ( z ) + ψ TagBox["\[Psi]", PolyGamma] ( a ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ) π z arg ( z ) 0 a + b 1 - a π ( ( a + b ) π z a + b - 1 ( 1 - a - b ) Β Beta ( a , b ) + csc ( ( a + b ) π ) sin ( b π ) ) arg ( z ) 0 - b π z a + b - 1 ( 1 - a - b ) Β Beta ( a , b ) + ( sin ( a π ) a π ) ( π - log ( z ) + ψ TagBox["\[Psi]", PolyGamma] ( a ) - ψ TagBox["\[Psi]", PolyGamma] ( a + b ) ) π arg ( z ) > 0 a + b - 1 + - a π sin ( a π ) ( a + π z + z ( - log ( z ) + ψ TagBox["\[Psi]", PolyGamma] ( a ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ) π z arg ( z ) > 0 a + b 1 - b π z a + b - 1 ( 1 - a - b ) Β Beta ( a , b ) + a π csc ( ( a + b ) π ) sin ( b π ) True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional BetaRegularized z a b -1 a a b z a b -1 1 -1 a -1 b Beta a b -1 -1 a z -1 PolyGamma a PolyGamma a b -1 z 0 a b -1 SuperPlus -1 -1 a a a -1 z z -1 z PolyGamma a z -1 z 0 a b 1 -1 a a b z a b -1 1 -1 a -1 b Beta a b -1 a b b z 0 -1 b z a b -1 1 -1 a -1 b Beta a b -1 a a -1 z PolyGamma a -1 PolyGamma a b -1 z 0 a b -1 SuperPlus -1 a a a z z -1 z PolyGamma a z -1 z 0 a b 1 -1 b z a b -1 1 -1 a -1 b Beta a b -1 a a b b Rule z [/itex]

 Rule Form

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

 2007-05-02