Inverse of the regularized incomplete gamma function: Differentiation (formula 06.12.20.0002)

InverseGammaRegularized

$Mathematica Notation$

Gamma, Beta, Erf

InverseGammaRegularized[a,z]

Differentiation

Low-order differentiation

With respect to a

http://functions.wolfram.com/06.12.20.0002.01

Input Form

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

Standard Form

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MathML Form

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Rule Form

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

2001-10-29

InverseGammaRegularized[a,z₁,z₂]