My question is about the relationship of the laplace transform and Fourier transform. I notice they are both very similar and besides the fact that the laplace transform is one sided(as opposed to the Fourier transform being two sided), the only other difference is that the Fourier transform has a purely imaginary component in its exponent while the laplace transform has a complex component.
My understanding is that the Fourier transform gives you a function of frequency that tells you which frequencies have the strongest amplitude in the original function of time. But since the laplace transform is kind of an extension of the Fourier transform that has this extra real term in its exponential, does that essentially mean the laplace transform is telling you the strongest frequencies AND the strongest exponential decays?(as opposed to only telling you information about the frequencies).
If this is the case, is there a corresponding laplace series you can write for a function that contains sines, cosines, and exponentials?