Exercise 21

Define the gamma function to be $$\Gamma(x) = \int_0^\infty e^{-t} t^{x-1} \; dt$$ for $x \gt 0$. The gamma function is very useful for expressing the Laplace transform of the function $t^\alpha$.

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

2. Prove that $\Gamma(x + 1) = x\Gamma(x)$ and deduce that $\Gamma(n+1) = n!$ for $n \in {\mathbb N}$.

3. Show that the Laplace transform of $f(t) = t^\alpha$ is $${\mathcal L}(f)(s) = \frac{\Gamma(\alpha + 1)}{s^{\alpha + 1}}.$$ If $n$ is a positive integer, then ${\mathcal L}(t^n)(s) = n!/s^{n + 1}$.