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variants of this functions
InverseBetaRegularized






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > InverseBetaRegularized[z,a,b] > Series representations > Generalized power series > Expansions at generic point a==a0 > For the function itself





http://functions.wolfram.com/06.23.06.0005.01









  


  










Input Form





InverseBetaRegularized[z, a, b] \[Proportional] InverseBetaRegularized[z, Subscript[a, 0], b] - (1 - w)^(1 - b) w ((-(1/Subscript[a, 0]^2)) HypergeometricPFQ[ {Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] + (Beta[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[ Subscript[a, 0] + b]))/w^Subscript[a, 0]) (a - Subscript[a, 0]) + (1/2) (1 - w)^(1 - 2 b) w (-2 (1 - w)^b Gamma[Subscript[a, 0]]^3 HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] - ((1 - w)^b Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (PolyGamma[Subscript[a, 0]] - PolyGamma[Subscript[a, 0] + b]) (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b]))/ w^Subscript[a, 0] + (-1 + b) w^(1 - Subscript[a, 0]) Gamma[Subscript[a, 0]]^2 HypergeometricPFQRegularized[ {Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] (w^Subscript[a, 0] Gamma[Subscript[a, 0]]^2 HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] - Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b])) + ((-1 + w) Gamma[Subscript[a, 0]]^2 (-HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] + Subscript[a, 0]^2 (-1 + b) w HypergeometricPFQRegularized[ {1 + Subscript[a, 0], 1 + Subscript[a, 0], 2 - b}, {2 + Subscript[a, 0], 2 + Subscript[a, 0]}, w]) (w^Subscript[a, 0] Gamma[Subscript[a, 0]]^2 HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] - Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b])))/w^Subscript[a, 0] + (1 - b) w^(1 - 2 Subscript[a, 0]) Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b]) (w^Subscript[a, 0] Gamma[Subscript[a, 0]]^2 HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] - Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b])) - ((1 - w)^b Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b]) (-Log[w] + ((1 - Subscript[a, 0]) (1 - w)^(1 - b) (w^Subscript[a, 0] Gamma[Subscript[a, 0]]^2 HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] - Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b])))/w^Subscript[a, 0]))/w^Subscript[a, 0] + (Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] ((-1 + w) w^Subscript[a, 0] Gamma[Subscript[a, 0]]^2 HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] - (-1 + w) Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b]) + (1 - w)^b w^Subscript[a, 0] (PolyGamma[1, Subscript[a, 0]] - PolyGamma[1, Subscript[a, 0] + b])))/w^(2 Subscript[a, 0])) (a - Subscript[a, 0])^2 + \[Ellipsis] /; (a -> Subscript[a, 0]) && w == InverseBetaRegularized[z, Subscript[a, 0], b]










Standard Form





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





2007-05-02





© 1998- Wolfram Research, Inc.