Recurrence relation for gamma function

Statement

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

$\mathrm{\Gamma }(z)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{z-1}{e}^{-t}dt$

Then, $\mathrm{\Gamma }$ satisfies the following functional equation:

$\mathrm{\Gamma }(z+1)=z\mathrm{\Gamma }(z)$

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

In fact, we use this functional equation to extend $\mathrm{\Gamma }$ 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.