[drjesusphd]

> I do not know how to properly motivate the Laplace transform

I feel like this is impossible without going to the complex plane. Like the author said, taking the inverse Laplace transform is no joke.

I feel like I never properly understood the Laplace transform until I learned about Landau damping. This is when waves exist, but are damped in a collisionless plasma. This damping is not disspiation and the energy does not get converted into heat. The usual way of presenting this is to show that if one Fourier transforms in time, you get the wrong answer. The fact that the system begins at a certain state, and is thus an initial value problem, needs to be respected.

[semi-extrinsic]

The Laplace transform is somehow the continuous analog of a Taylor series expansion. You don't need complex analysis to motivate it. I sketched this for my students when TAing once upon a time, heavily inspired by this [1] nice MIT lecture.

I'm very surprised this isn't standard material. It makes the parallel between Laplace and Fourier transforms so much more intuitive, because you get Taylor series as a parallel to Fourier series.

[1] https://youtu.be/zvbdoSeGAgI

[nimish]

expand an analytic function in its taylor series then find its values on the unit circle. There's a Fourier series.

But the Fourier series uses global data than the taylor series which uses point data so they aren't perfect analogs.

A laplace transform is a fourier transform rotated in the complex plane (more or less), and if you allow the transform to take complex "frequencies" then they are basically unified. The difference is that the laplace transform is all about causal functions of time (t<0 => f(t) = 0) where as the fourier transform is less picky.