Derivative of Gamma Function

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Theorem

$\ds \map {\Gamma'} x = \int_0^\infty t^{x - 1} \ln t \, e^{-t} \rd t$

where $\map {\Gamma'} x$ denotes the derivative of the Gamma function evaluated at $x$.


Proof

\(\ds \map {\Gamma'} x\) \(=\) \(\ds \frac \d {\d x} \int_0^\infty t^{x - 1} e^{-t} \rd t\) Definition of Gamma Function
\(\ds \) \(=\) \(\ds \int_0^\infty \frac {\partial} {\partial x} t^{x - 1} e^{-t} \rd t\) Leibniz's Integral Rule
\(\ds \) \(=\) \(\ds \int_0^\infty t^{x-1} \ln t \, e^{-t} \rd t\) Derivative of Power of Constant

$\blacksquare$