Gauss' formula for gamma function

Theorem

The following formula holds for $z \in \mathbb{C} \setminus \{0,-1,-2,\ldots\}$: $$\Gamma(z) = \displaystyle\lim_{n \rightarrow \infty} \dfrac{n! n^z}{z(z+1)\ldots(z+n)},$$ where $\Gamma$ denotes the gamma function and $n!$ denotes the factorial.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 6.1.2