Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 1 / 14C / Solution

The convolution is $(f\ast g)(x)={\int }_{-\infty }^{\infty }f(y)g(x-y)dy$. Under the usual integrability hypotheses, Fubini's theorem and $u=x-y$ give

The supplied transform and the Fourier shift theorem giveSincethe shift and differentiation rules yieldThe coincident boundary values make the endpoint convention immaterial.