Gauss Multiplication Formula

Theorem
Let $\Gamma$ denote the Gamma Function.

Then:


 * $\displaystyle \forall z \notin \left\{{-\frac m n: m \in \N}\right\}: \prod_{k \mathop = 0}^{n - 1} \Gamma \left({z + \frac k n}\right) = \left({2 \pi}\right)^{\left({n - 1}\right) / 2} n^{1/2 - n z} \Gamma \left({n z}\right)$

where $\N$ denotes the natural numbers.

Proof
Taking the product for $k = 0$ to $n - 1$, we have: