Logarithm of the gamma function: Differentiation (formula 06.11.20.0006)

LogGamma

$Mathematica Notation$

Gamma, Beta, Erf

LogGamma[z]

Differentiation

Fractional integro-differentiation

http://functions.wolfram.com/06.11.20.0006.01

Input Form

D[LogGamma[a z + b], {z, \[Alpha]}] == Sum[(a z)/(k Gamma[2 - \[Alpha]]) - ((a z)/(b + k)) Hypergeometric2F1Regularized[1, 1, 2 - \[Alpha], -((a z)/(b + k))] + (b/k - Log[(b + k)/k])/Gamma[1 - \[Alpha]], {k, 1, Infinity}]/ z^\[Alpha] - (a EulerGamma z^(1 - \[Alpha]))/Gamma[2 - \[Alpha]] - ((a z^(1 - \[Alpha]))/b) Hypergeometric2F1Regularized[1, 1, 2 - \[Alpha], -((a z)/b)] - (Log[b] + EulerGamma b)/(z^\[Alpha] Gamma[1 - \[Alpha]])

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