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Breaking functions down to their constituent parts is nice and all, but this is a step too far.
(fedia.io)
this post was submitted on 29 Nov 2025
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Explanation: Top left is a Taylor series, which expresses an infinitely differentiable function as an infinite polynomial. Center left is a Fourier transform, which extracts from periodic function into the frequencies of the sines and cosines composing it. Bottom left is the Laplace transform, which does the same but for all exponentials (sines and cosines are actually exponentials, long story). It seems simpler than the Fourier transform, until you realize that the s is a complex number. In all of these the idea is to break down a function into its component parts, whether as powers of x, sines and cosines or complex exponentials.
Edit: I'll try to explain if something is unclear, but... uh... better not get your hopes up.
Oh, look at that hornet's nest. I wonder what happens if I poke it
As someone who worked with system modelling, analysis and control for years... I do think the Laplace transform is easier to work with ๐๐โโ๏ธ
It is, but conceptually it's a lot weirder than the Fourier transform, whose idea at least is very straightforward. I mean, when doing Laplace transforms you do have to assume that int(e^tdt){0}{โ}=-1. I'd definitely rather use the Laplace transform, but you couldn't pay me to explain how that shit actually works to an undergrad student.
Basically the assumption is that the signal x(t) is equal to 0 for all t < 0 and that the integral converges. And what is a bit counter-intuitive: Laplace transformations can be regarded as generalizations of Fourier transformations, since the variable s is not only imaginary but fully complex. But yeah... I would have to brush up on it again, before explaining it as well. It's... been a while.