Legendre's Duplication Formula/Proof 1

Proof
From the definition of the Beta function:

Letting $z_1 = z_2 = z$ gives:

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


 * $\displaystyle \map \Beta {z_1, z_2} = \int_0^1 x^{2 z_1 - 2} \paren {1 - x^2}^{z_2 - 1} 2 x \rd x$

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


 * $(2): \quad \displaystyle \map \Beta {\frac 1 2, z} = 2 \int_0^1 \paren {1 - x^2}^{z - 1} \rd x$

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

From Gamma Function of One Half:
 * $\map \Gamma {\dfrac 1 2} = \sqrt \pi$

It follows that:


 * $\map \Gamma z \, \map \Gamma {z + \dfrac 1 2} = 2^{1 - 2 z} \sqrt \pi \, \map \Gamma {2 z}$