Chapter 5.1, Problem 36P

The Gamma Function. The gamma function is defined by $\Gamma (p)$ and defined by the integral

$\Gamma (p+1)={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-x}{x}^p}dx$.         (i)

The integral converges for all $p>-1$.

(a) Show that, for $p>0$,

$\Gamma (p+1)=p\Gamma (p)$

(b) Show that $\Gamma (1)=1$.

(c) If $p$ is a positive integer $n$, show that

$\Gamma (n+1)=n!$.

Since $\Gamma (p)$ is also defined when $p$ is not an integer, this function provides an extension of the factorial function to nonintegral values of the independent variable. Note that it is also consistent to define $0!=1$.

(d) Show that, for $p>0$,

$p(p+1)\cdots (p+n-1)=\frac{\Gamma (p+n)}{\Gamma (p)}$.

Thus $\Gamma (p)$ can be determined for all positive values of $p$ if $\Gamma (p)$ is known in a single interval of unit length, say, $0<p\le 1$. It is possible to show that $\Gamma (1/2)=\sqrt{\pi }$. Find $\Gamma (3/2)$ and $\Gamma (11/2)$.