For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show that $\Gamma^{\prime}$ vanishes at a unique real number $\xi$ whose integer part will be determined.
For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show that $\Gamma^{\prime}$ vanishes at a unique real number $\xi$ whose integer part will be determined.