There is a famous theorem says about Gamma function:
Bohr-Mollerup Theorem
Let $f:(0,\infty)\rightarrow \mathbb{R}^+$ be a function satisfying below
(i) $f(x+1)=xf(x), \forall x\in (0,\infty)$
(ii) $f$ is a log-convex function
(iii) $f(1)=1$
Then, $f=\Gamma$ on its domain
This theorem only shows that the Gamma function deserves to be called the factorial function for $x\in\mathbb{R}^+$.
Is it possible to extend this idea to complex plane so Gamma function deserves to be the best factorial function?
Since $t^{z-1}e^{-t}$ can be decomposed into its real part and imaginary part using cosine&sine function, i guess it would show that the Gamma function on complex plane has some flexible property similar to that on real line.