I am currently Reading about laplace transforms and their use in linear ODE's with constant coefficients,and I have some understanding that why the" Fourier series" relates to frequency/time transformations,but I am unable to draw the conclusion in the laplace case.
Wikipedia only vaguely explains this ,so a understandable explanation about how that works (and easy to follow)would be appreciated .
Also ,a mini question is that why don't we define a faster growing function (like $e^{t^2}$),so that for a wider range of functions the improper integral would converge?