TITLE: Gamma function proof of gamma $\;Γ(1/2) = \sqrt \pi\;$
QUESTION [7 upvotes]: So our teacher doesnt use the same demonstration as most other sites use for proving that gamma of a half is the square root of pi.
I dont understand the demonstration from the first step because he uses the Wallis product but first he represents $Γ(1/2)$ as : 
$$Γ(n + 1/2) = 2^{-n}Γ(1/2)\prod_{k=1..n}(2k-1)$$
This is just the first step and i dont undderstand how they get that..
I understand the gamme function and that when you integrate it you get $Γ(x+1) = xΓ(x)$ and i know i need to somehow use this identity but i dunno how.

REPLY [2 votes]: We use Euler's Reflection Formula $\Gamma(1-z)\Gamma(z)=  \frac\pi{\sin(\pi z)}$ with $z=\frac12$.
We then have $\Gamma(1/2)^2=\frac\pi{\sin( \frac\pi2)}$.
Since $\sin( \frac\pi2)=1$ we now have $\Gamma(1/2)^2=\pi$. Finally therefore $\Gamma(1/2)=\sqrt\pi$. Not $-\sqrt\pi$ because we know $\Gamma( 1/2) > 0$.