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First of all, I just don't understand what makes the Laplace transform work so well. All of my books on differential equations just give the definition and then results and such. But how is the integral derived? I shudder to think that someone just "noticed" it works.

On to Fourier.. I have not formally studied the Fourier transform yet, but I have studied the Fourier series. My first reaction to seeing the transform is that is looks so similar to the Laplace transform that maybe the Laplace is a specific case of the Fourier transform (kind of doubt that). I am still not sure exactly how the Fourier transform comes out of the Fourier series. How does this work? There seems to be a resemblance between the series in exponential form and the integral definition, but I am still not sure how the series goes into the integral definition.

Does the Fourier transform work pretty much the same way as the Laplace transform only it takes care of complex functions?

Also, are these two things linked somehow? Or should I stop with the bad habit of assumption and start to think of these as two completely independent things?

Sorry if this type of thing has been posted before. I used the search option but didn't find much.