Factorial....

Factorial of any real number $n$ is defined by Gamma function as follows:

$\Gamma (n) = (n-1)!$

$\quad \Rightarrow ( \dfrac{1}{2} )! ( -\dfrac{1}{2} ) ! = ( \dfrac{3}{2}-1 ) ! ( \dfrac{1}{2}-1 ) ! = \Gamma ( \dfrac {3} {2} ) \Gamma ( \dfrac {1}{2} )$

It is also known that:

$\Gamma {(1+z)} = z\Gamma {(z)}$

$\quad \Rightarrow \Gamma ( \dfrac {3} {2} ) \Gamma ( \dfrac {1}{2} ) = \dfrac {1}{2} \Gamma ( \dfrac {1}{2} )\Gamma ( \dfrac {1}{2} ) = \dfrac {1}{2} \left( \Gamma ( \dfrac {1}{2} ) \right)^2$

Since $\Gamma ( \dfrac {1}{2} ) = \sqrt{\pi}$ , then we have:

$\quad \Rightarrow (\dfrac{1}{2} )! ( -\dfrac{1}{2} ) ! = \dfrac {1}{2} \left( \Gamma ( \dfrac {1}{2} ) \right)^2 = \boxed {\dfrac {\pi} {2} }$

same solution as chew. So whenever face factorial of rational number we can think of gamma functions to solve that problem