Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2007 / 9 / ii / Solution

For $\mathrm{Re}p>0$, the starting integral is ${\int }_0^{\infty }{e}^{-pt}dt=1/p$. Repeated integration by parts, with vanishing boundary terms, givesThis also follows from the gamma function. The stated open half-plane ensures absolute convergence.