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

For $\mathrm{Re}p>a$, differentiation under the integral is justified by the dominated convergence theorem: on a smaller closed half-plane $\mathrm{Re}p\ge a+\delta$, the additional factor $t$ is controlled by $t{e}^{-\delta t}$. HenceThis is the multiplication rule for the Laplace transform.