Recurrence relation for gamma function

Statement

Let $\Gamma$ denote the gamma function as defined on the right half-plane by:

$\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt$

Then, $\Gamma$ satisfies the following functional equation:

$\Gamma (z+1)=z\Gamma (z)$

whenever $z$ is in the right half-plane.

In fact, we use this functional equation to extend $\Gamma$ to a meromorphic function on $\mathbb {C}$, so the functional equation holds more generally for any $z$ that is not a non-positive integer (non-positive integers are precisely the simple poles).

Proof

The proof is using integration by parts.