analysis - Why is $Gammaleft(frac{1}{2}right)=sqrt{pi}$ ?

It seems as if no one has asked this here before, unless I don't know how to search.

The Gamma function is
$$\Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx.$$
Why is
$$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\ ?$$
(I'll post my own answer, but I know there are many ways to show this, so post your own!)