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 |
$\blacksquare$