Gamma Difference Equation/Proof 1

Theorem
The Gamma function satisfies:


 * $\Gamma \left({z+1}\right) = z \Gamma \left({z}\right)$

for all $z \in \C$ which is not a nonpositive integer.

Proof
Let $z \in \C$, with $\operatorname {Re} \left({z}\right) > 0$. Then

If $z \in \C \setminus \left\{{0, -1, -2, \ldots}\right\}$ such that $\operatorname {Re} \left({z}\right) \le 0$, then the statement holds by the definition of $\Gamma$ in this region.