stumbled upon it years ago but I’m terrible at math so kinda forgot about it for years, recently stumbled upon it again so out of pure curiosity, what is it? does it have real world use or is it something purely theoretical?
In: Mathematics
The Laplace Transform is “just” a disguised Fourier transform. They both do the same sort of thing: treat the space of functions as an infinite-dimensional vector space, and change to a different set of basis vectors. The new basis vectors happen to diagonalize the matrices that correspond to differentiation, I.e. they convert differentiation to multiplication (and vice versa). The Fourier transform uses oscillating functions (imaginary exponentials) as a basis for all functions, and the Laplace transform instead uses real exponentials. The two are exactly equivalent if you allow the frequency (of the Fourier transform) or the *s* parameter (of the Laplace transform) to be complex.
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