Chapter 5.8, Problem 13P

(a) If $f(t)=t^m$ and $g(t)=t^n$, where $m$ and $n$ are positive integers, show that

$\left(f\ast g\right)\left(t\right)=t^{m+n+1}{\displaystyle \underset{0}{\overset{1}{\int }}{u}^m{\left(1-u\right)}^{n}du}$.

(b) Use the convolution theorem to show that

${\displaystyle \underset{0}{\overset{1}{\int }}{u}^m{\left(1-u\right)}^{n}du}=\frac{m!n!}{\left(m+n+1\right)!}$

(c) Extend the result of part (b) to the case where $m$ and $n$ are positive numbers but not necessarily integers.