Legendre's Duplication Formula

Theorem
Let $\Gamma$ denote the gamma function.

Then:
 * $\displaystyle \forall z \notin \left\{ -\frac n 2 : n \in \N_0 \right\}: \Gamma \left({z}\right) \Gamma \left (z + \frac 1 2 \right) = 2^{1-2z} \sqrt \pi \Gamma \left({2 z}\right)$

where $N_0 = \N \cup \left\{{0}\right\}$.

Proof
From the Beta Function we have:

Letting $z_1 = z_2 = z$ gives:

Now substituting $u = x^2$ into the Beta function:


 * $\displaystyle B \left({z_1, z_2}\right) = \int_0^1 x^{2z_1 - 2} \left({1 - x^2}\right)^{z_2 - 1} 2x \ \mathrm d x$

Letting $z_1 = \dfrac 1 2$ and $z_2 = z$ gives:


 * $(2): \quad \displaystyle B \left({\frac 1 2, z}\right) = 2 \int_0^1 \left({1 - x^2}\right)^{z - 1} \ \mathrm d x$

Combining results $(1)$ and $(2)$:

Letting $z = \dfrac 1 2$ in Euler's Reflection Formula we have:

It follows that:


 * $\Gamma \left({z}\right) \Gamma \left({z + \dfrac 1 2}\right) = 2^{1 - 2z} \sqrt \pi \Gamma \left({2z}\right)$