Hankel contour integral for the gamma function

Remarks

The integral converges for every $z$ because $e^t$ decays fast as $\operatorname{Re} t \to -\infty$, and it is an entire function of $z$. For $\operatorname{Re} z < 1$, the path can be squeezed onto the cut, with $t = s e^{-i\pi}$ on the lower side and $t = s e^{i\pi}$ on the upper side, $s > 0$. The small circle around $0$ contributes nothing in the limit. The two sides give $\frac{1}{2\pi i}(e^{i\pi z} - e^{-i\pi z}) \int_0^\infty e^{-s} s^{-z}\, ds = \frac{\sin(\pi z)}{\pi}\,\Gamma(1 - z)$, which equals $1/\Gamma(z)$ by the reflection formula. The identity theorem then extends the formula to all $z \in \C$. At a positive integer $z = n$ the integrand $e^t t^{-n}$ is single-valued, and the formula reduces to the residue at $0$, which is $1/(n-1)!$.

Going deeper

For a student

The Hankel integral gives a direct proof that $1/\Gamma$ is entire, without first building the meromorphic continuation of $\Gamma$. Similar loop integrals represent other functions with branch points, such as the Riemann zeta function, and the same technique gives their continuations. It is also a starting point for asymptotic estimates of $1/\Gamma$ by the saddle point method.