Laplace transform of a squared function

Question

The Laplace transform often comes in handy for solving linear ODEs. But, linear ODEs are often approximations of nonlinear ODEs that have less trivial solutions.

Given $f(t)$ and that its laplace transform exists, is there a closed formula for the laplace transform of $f(t)^2,$ and more generally,$ f(t)^n?$

Answer 1

Yes, the Laplace transform of $f(t)^2$ is $(F\star F)(s)$ where $\star$ denotes convolution. Similarly, the Laplace transform of $f(t)^n$ is given by $(\underbrace{F\star \ldots\star F}_{n\ \mathrm{ times}})(s)$.