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While messing around trying to calculate a power series for the Gamma function, I ran across this integral:
$$\int_0^\infty(\log t)^n e^{-t}\ dt,\ n \in \mathbb{N}$$
I've looked at it for a while and tried a couple of things, but I'm stumped. Is there a way to calculate this?
It is the $n$th derivative of the gamma function evaluated at the point $s=1$, where gamma function is given by
$$ \Gamma(s)=\int_{0}^{\infty} x^{s-1}e^{-x} dx \implies \Gamma^{(n)}(s)|_{s=1}=\int_{0}^{\infty} (\ln(x))^{n}e^{-x} dx .$$
Added: You can start from the point
$$ \psi(x) = \frac{d}{dx}\ln \Gamma(x) =\frac{\Gamma'(x)}{\Gamma(x)} \implies \Gamma'(x)=\Gamma(x)\psi(x),$$
where $\psi(x)$ is the digamma function.
See the pattern? $$ \int_{0}^{\infty} \operatorname{ln} (t)^{7} \operatorname{e} ^{-t} d t = -\gamma^{7} - \frac{61 \pi^{6} \gamma}{24} - 84 \zeta (5) \pi^{2} - \frac{21 \pi^{4} \zeta (3)}{2} - 280 \zeta (3)^{2} \gamma - 70 \zeta (3) \pi^{2} \gamma^{2} - 504 \zeta (5) \gamma^{2} - \frac{21 \pi^{4} \gamma^{3}}{4} - 70 \zeta (3) \gamma^{4} - \frac{7 \pi^{2} \gamma^{5}}{2} - 720 \zeta (7) $$ (Computed by Maple)
