Gauss Multiplication Formula

Theorem
Let $\Gamma$ denote the Gamma Function.

Then:


 * $\ds \forall z \notin \set {-\frac m n: m \in \N}: \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1/2 - n z} \map \Gamma {n z}$

where $\N$ denotes the natural numbers.

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