Chapter 6
The Laplace Transform

Section 5.9 obtained the Fourier transform by letting the interval grow without bound, and observed that differentiation becomes multiplication under it. That is the property worth having, and it is what makes a transform useful: an operation that is hard in one representation becomes easy in the other.

The Fourier transform has two drawbacks in practice. It requires \(f\) to be absolutely integrable over the whole line, which excludes functions as ordinary as \(f(t)=1\) or \(f(t)=e^{t}\); and it takes no account of initial conditions, which is precisely what one has when solving an equation forward in time. The Laplace transform repairs both. It integrates only over \(t\geq 0\), so the past is not required, and it inserts a decaying factor \(e^{-st}\) that tames growth. The price is that the transform variable becomes complex and the inversion is less symmetric.