I need to prove that the characteristic function of the gamma distribution with parameters $(r,\lambda)$ is $$ \phi(t)=(\lambda/(\lambda-it))^r$$
I was told to prove, using complex integration, that $$\int_0^{\infty} \alpha^rx^{r-1}e^{-\alpha x} dx= \Gamma(r)$$ if $Re(\alpha) >0$.
I tried to rewrite the integral, looking at it as an integral in the complez variable $z=\alpha x$ but i did not manage to end the proof.
In particular, i tried to consider the complex integral on the path that is the angle between the real line and the $\alpha$ line but i wasn't able to proof that the integral on the arch connecting them is $0$.