Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2012 / 9 / iii / Solution

Write $I_n(p)={\int }_0^{\infty }{e}^{-pt}{t}^{n}dt$. For $\mathrm{Re}p>0$, exponential decay dominates every fixed power of $t$, so the improper integral converges absolutely and $e^{-pT}{T}^n\to 0$. For $n\ge 1$, integration by parts givesThe preceding part gives the base case $I_0(p)=1/p$. Repeated application, or mathematical induction, yields