If you are interested into some additional insights into the Laplace transform (and its connection to the Fourier transform), consider watching one (or both) of the videos below. They both give some nice visualizations! (Note: In the video, they use a very slightly different definition for their Fourier transform. This is an issue that often crops up!)
MajorPrep/Zach Star from these videos is a Cal Poly EE grad! I wish more EE students got to take and explore MATH classes like MATH 344.