Chapter 5.1, Problem 37P

Consider the laplace transform of $t^p$, where $p>-1$.

(a) Referring to problem $36$, Show that

$\begin{array}{l}L\left\{t^p\right\}={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-st}{t}^p}dt=\frac{1}{s^{p+1}}{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-x}{x}^p}dx\\ \\ =\frac{\Gamma (p+1)}{s^{p+1}},s>0.\end{array}$

(b) Let $p$ be a positive integer $n$ in (a). Show that

$L\left\{t^n\right\}=\frac{n!}{s^{n+1}},s>0$

(c) Show that

$L\left\{t^{-1/2}\right\}=\frac{2}{\sqrt{s}}{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-x^2}}dx,s>0$.

It is possible to show that

${\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-x^2}}dx=\frac{\sqrt{\pi }}{2}$;

Hence,

$L\left\{t^{-1/2}\right\}=\sqrt{\frac{\pi }{s}},s>0$.

(d) Show that

$L\left\{t^{1/2}\right\}=\frac{\sqrt{\pi }}{2s^{3/2}},s>0$.

36. 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 $p>-1$.

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

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

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

(g) 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 $0!=1$.

(h) 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 $\Gamma (11/2)$.