Legendre's Duplication Formula

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.

Also denoted as
Some sources report this as:
 * $\forall z \notin \left\{{-\dfrac n 2: n \in \N}\right\}: 2^{2 z - 1} \Gamma \left({z}\right) \Gamma \left (z + \dfrac 1 2 \right) = \sqrt \pi \ \Gamma \left({2 z}\right)$