Chapter 11.6, Problem 7P

(a) Given that $\Gamma \left(1/2\right)=\sqrt{\pi }$ by Problem 6, determine $\Gamma \left(3/2\right)$ and $\Gamma \left(-1/2\right)$.

(b) Show that for positive integer $n$:

$\Gamma \left(n+\frac{1}{2}\right)=\frac{\left(2n\right)!}{2^{2n}\cdot n!}\sqrt{\pi }.$

(c) Show that for positive integer $n$:

$\Gamma \left(\frac{1}{2}-n\right)=\frac{{\left(-1\right)}^n\cdot 2^{2n}\cdot n!}{\left(2n\right)!}\sqrt{\pi }.$