Laplace transform of distributions

Summary

The Laplace transform extends from functions to distributions supported in $[0, \infty)$ that do not grow faster than an exponential. The extension gives meaning to transforms of generalized functions such as the Dirac delta and its derivatives: $\mathcal{L}\delta = 1$ and $\mathcal{L}\delta^{(k)} = s^k$. It makes the rule "differentiation becomes multiplication by $s$" exact, with no initial-value terms, which is the natural setting for impulses in operational calculus.