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.
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 🙈🏃♂️