Breaking functions down to their constituent parts is nice and all, but this is a step too far.
Breaking functions down to their constituent parts is nice and all, but this is a step too far.
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They should have cancelled this series instead of Rome and Firefly.
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Taylor... You fear to go into those series. Euler delved too greedily and too deep. You know what they awoke in the depths of the number line...
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Wat?
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Added an explanation comment. Should've probably done that sooner.
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Ah yes.... it's crystal clear now!
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14 comments
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 🙈🏃♂️
What kind of work do you do?
I'm in the process of wrapping up my degree and I work a lot with signals and controls. I agree that Laplace is much less of a headache than Fourier.
Can you elaborate on why without getting us all stung to death?
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}{∞}=-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.
I understand some of these words