Gamma Difference Equation

Theorem
The Gamma function satisfies


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

for any $z \ $ which is not a nonpositive integer.

Proof 1
Let $z \in \C$, with $\Re(z) > 0$. Then

If $z \in \C \backslash \{0,-1,-2,\ldots\}$ with $\Re(z) \leq 0$, then the statement holds by the definition of $\Gamma$ in this region.

Proof 2
By Euler's form of the Gamma function,