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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 b==b0 > For the function itself





http://functions.wolfram.com/06.23.06.0007.01









  


  










Input Form





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










Standard Form





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MathML Form







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





2007-05-02





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