Legendre's Duplication Formula/Proof 2

Proof
From Gauss Multiplication Formula:


 * $\ds \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1/2 - n z} \map \Gamma {n z}$

Substituting $n = 2$ yields: