# Euler-Gauss Limit Formula for the Complex Gamma Function

Suppose we have the function $$\Gamma$$ defined for all $$z \in \mathbb{C}$$ with $$\mathrm{Re}~ z > 0$$ by the standard integral formula $$\Gamma(z) = \int_{0}^{\infty} e^{-t} t^{z - 1} dt$$ (perhaps extended to $$\mathbb{C}$$ except $$0,-1,-2,\ldots$$ via recurrent relation), and that we aim to prove the identity $$\Gamma(z) = \lim_{n \to \infty} \frac{n^z n!}{z (z+1) \ldots (z + n)}.$$ In the real case, there is a relatively simple proof of this identity utilising log-convexity of $$\Gamma$$ (demonstrated, e.g., in Artin's Gamma Function).

Is there any simple way how to use the identity for real-variable $$\Gamma$$ (or the log-convexity argument) to obtain the identity in the complex case? It is trivial if one knows that both sides of the identity are analytic functions – but this seems to me as relatively non-trivial compared to the rest of the stuff.

My main motivation for asking this question is that I have found multiple sources, which effectively state the proof for $$\Gamma$$ over $$\mathbb{R}$$ and then claim that they have proved the property for $$\Gamma$$ over $$\mathbb{C}$$. I am therefore curious if there is some simple argument that can be used in the transtion from $$\mathbb{R}$$ to $$\mathbb{C}$$.