Lecture slides: the-laplace-transform.pdf

The Laplace transform is central to the analysis of input-output systems. Here we learn the basics, and introduce the concept of the region of convergence, and analytic continuation.

A nice extra reference for the Laplace Transform is these notes - all credit to Haynes Miller and Jeremy Orloff (MIT):

laplace-notes.pdf

3blue1brown gives an excellent introduction to analytic continuation in this video on the Riemann zeta function (extra credit for any student that can solve the Riemann Hypothesis).