Past exam of the short options course of the University of Oxford / 2015 / 5 / a

Use the unilateral Laplace transform convention $\overline{f}(p)={\int }_0^{\infty }{e}^{-px}f(x)dx$. The Laplace transform of a derivative isThe formula holds in the common Laplace convergence region under the usual endpoint and growth hypotheses. The initial value is the right-hand value at zero.

With the same unilateral Laplace transform convention, the standard sine formula is

The defining evaluation rule of the Dirac delta givesThe impulse lies strictly inside the integration domain, so there is no endpoint convention to settle. This Laplace transform exists for every complex $p$.