Chapter 3: Laplace Transforms

Every input in the last chapter was a smooth, everlasting sinusoid. Real machinery is switched: a relay closes, a thermostat clicks, a load lands on a beam at one instant and leaves at another. At the moment of switching the forcing jumps, and the methods built so far turn awkward — undetermined coefficients wants a single formula valid for all time, and a signal that is "nothing, then suddenly something" refuses to be one formula. This chapter's instrument, the Laplace transform $F(s)={\int }_0^{\infty }{e}^{-st}f(t)dt$, is built for exactly this world: it integrates the entire history of a signal, jumps and all, into one well-behaved function of a new variable $s$.