Why is \Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\Gamma\left(\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}\text{ ?}

(I’ll post my own answer, but I know there are many ways to show this, so post your own!)

Answer

We only need Euler’s formula:

\Gamma(1-z) \Gamma(z) = \frac{\pi}{\sin \pi z} \Longrightarrow \Gamma^2\left(\frac{1}{2}\right ) = \pi

Attribution
Source : Link , Question Author : Michael Hardy , Answer Author : DonAntonio

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