Value of gamma function $\Gamma\left(\frac12-n\right)$

Question

In Wikipedia it is claimed that for nonnegative integer $n$,

$$\Gamma\left(\frac12-n\right)=\dfrac{(-4)^nn!}{(2n)!}\sqrt{\pi}.$$

How to prove that?

Answer 1

You first find $\Gamma(1/2),$ which is equivalent to evaluating $\int_\infty^\infty e^{-x^2} dx,$ and then you prove the functional equation for $\Gamma,$ which says that $\Gamma(x+1)= x \Gamma(x).$ The last can be done by writing down the definition of $\Gamma$ and integrating by parts.