Legendre's Duplication Formula/Proof 2

Theorem
Let $\Gamma$ denote the gamma function.

Then:


 * $\forall z \notin \left\{{-\dfrac n 2: n \in \N}\right\}: \Gamma \left({z}\right) \Gamma \left (z + \dfrac 1 2 \right) = 2^{1 - 2 z} \sqrt \pi \, \Gamma \left({2 z}\right)$

where $\N$ denotes the natural numbers.

Proof
From Gauss Multiplication Formula:


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

Substituting $n=2$ yields: