Legendre's Duplication Formula/Proof 2

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: