Properties of gamma function What is the simplest way to prove the two following properties of $
\mathit{\Gamma}{\mathrm{(}}{z}{\mathrm{)}}
$
a) 
$
\mathit{\Gamma}{\mathrm{(}}\frac{3}{2}{\mathrm{)}}\mathrm{{=}}\frac{1}{2}\mathit{\Gamma}{\mathrm{(}}\frac{1}{2}{\mathrm{)}}\mathrm{{=}}\frac{\sqrt{\mathit{\pi}}}{2}
$
b) $
\mathit{\Gamma}{\mathrm{(}}{z}{\mathrm{)}}\mathit{\Gamma}{\mathrm{(}}{1}\mathrm{{-}}{z}{\mathrm{)}}\mathrm{{=}}\frac{\mathit{\pi}}{\sin{\mathrm{(}}\mathit{\pi}{z}{\mathrm{)}}}
$
Thanks very much in advance.

For the first property we are looking for :
$\Gamma (z+1)=z\Gamma(z)$ which was introduced by Gauss and is sometimes called $\Pi (z)$.
This can be derived from the integral representation of Gamma.
The second property is called Euler' reflection formula. The proof is available here : 
https://proofwiki.org/wiki/Euler's_Reflection_Formula#Proof
