Inversion of rational transforms by partial fractions

Summary

A rational function $F(s) = P(s)/Q(s)$ with $\deg P < \deg Q$ is the Laplace transform of a sum of terms $t^{k-1} e^{a t}$. To find the inverse Laplace transform, one splits $F$ by partial fraction decomposition into terms $c/(s - a)^k$ and inverts each term from the rule $\mathcal{L}\{t^{k-1} e^{a t}/(k-1)!\} = 1/(s - a)^k$. This is the everyday method for the transforms that arise from linear differential equations with constant coefficients.