The Euler Gamma function is defined, for all real $x > 0$, by: $$\Gamma(x) = \int_{0}^{+\infty} e^{-t} t^{x-1} dt$$ Justify that the function $\Gamma$ is of class $\mathcal{C}^{1}$ and strictly positive on $]0, +\infty[$.
The Euler Gamma function is defined, for all real $x > 0$, by:
$$\Gamma(x) = \int_{0}^{+\infty} e^{-t} t^{x-1} dt$$
Justify that the function $\Gamma$ is of class $\mathcal{C}^{1}$ and strictly positive on $]0, +\infty[$.