Inverse of the regularized incomplete beta function: Differentiation (formula 06.23.20.0004)

InverseBetaRegularized

$Mathematica Notation$

Gamma, Beta, Erf

InverseBetaRegularized[z,a,b]

Differentiation

Low-order differentiation

With respect to a

http://functions.wolfram.com/06.23.20.0004.01

Input Form

D[InverseBetaRegularized[z, a, b], {a, 2}] == (1 - w)^(1 - 2 b) w (-2 (1 - w)^b Gamma[a]^3 HypergeometricPFQRegularized[ {a, a, a, 1 - b}, {1 + a, 1 + a, 1 + a}, w] - ((1 - w)^b Beta[a, b] BetaRegularized[w, a, b] (PolyGamma[a] - PolyGamma[a + b]) (Log[w] - PolyGamma[a] + PolyGamma[a + b]))/w^a + (-1 + b) w^(1 - a) Gamma[a]^2 HypergeometricPFQRegularized[{a, a, 1 - b}, {1 + a, 1 + a}, w] (w^a Gamma[a]^2 HypergeometricPFQRegularized[{a, a, 1 - b}, {1 + a, 1 + a}, w] - Beta[a, b] BetaRegularized[w, a, b] (Log[w] - PolyGamma[a] + PolyGamma[a + b])) + ((-1 + w) Gamma[a]^2 (-HypergeometricPFQRegularized[{a, a, 1 - b}, {1 + a, 1 + a}, w] + a^2 (-1 + b) w HypergeometricPFQRegularized[ {1 + a, 1 + a, 2 - b}, {2 + a, 2 + a}, w]) (w^a Gamma[a]^2 HypergeometricPFQRegularized[{a, a, 1 - b}, {1 + a, 1 + a}, w] - Beta[a, b] BetaRegularized[w, a, b] (Log[w] - PolyGamma[a] + PolyGamma[a + b])))/w^a + (1 - b) w^(1 - 2 a) Beta[a, b] BetaRegularized[w, a, b] (Log[w] - PolyGamma[a] + PolyGamma[a + b]) (w^a Gamma[a]^2 HypergeometricPFQRegularized[{a, a, 1 - b}, {1 + a, 1 + a}, w] - Beta[a, b] BetaRegularized[w, a, b] (Log[w] - PolyGamma[a] + PolyGamma[a + b])) - ((1 - w)^b Beta[a, b] BetaRegularized[w, a, b] (Log[w] - PolyGamma[a] + PolyGamma[a + b]) (-Log[w] + ((1 - a) (1 - w)^(1 - b) (w^a Gamma[a]^2 HypergeometricPFQRegularized[{a, a, 1 - b}, {1 + a, 1 + a}, w] - Beta[a, b] BetaRegularized[w, a, b] (Log[w] - PolyGamma[a] + PolyGamma[a + b])))/w^a))/w^a + (Beta[a, b] BetaRegularized[w, a, b] ((-1 + w) w^a Gamma[a]^2 HypergeometricPFQRegularized[{a, a, 1 - b}, {1 + a, 1 + a}, w] - (-1 + w) Beta[a, b] BetaRegularized[w, a, b] (Log[w] - PolyGamma[a] + PolyGamma[a + b]) + (1 - w)^b w^a (PolyGamma[1, a] - PolyGamma[1, a + b])))/w^(2 a)) /; w == InverseBetaRegularized[z, a, b]

Standard Form

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MathML Form

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

2001-10-29

InverseBetaRegularized[z₁,z₂,a,b]