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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
Can you elaborate on why without getting us all stung to death?
Basically two things: 1. Complicated operations in the time domain like convolutions become simple operations in the frequency domain. 2. It is way easier to handle complex numbers and do analysis with them than with explicit frequencies to the point where some things like stability and robustness can be judged by simple geometry (e.g. are the eigenvalues within the unit circle) or the sign of the imaginary part.
EDIT: I forgot the most important simplification of operations: A derivative in the time domain, becomes a simple multiplication by s in the frequency domain. So solving Differential Equations of the system's dynamics becomes pretty easy without having to go back into the time domain explicitly.