TITLE: Fourier vs Laplace transforms
QUESTION [39 upvotes]: In solving a linear system, when would I use a Fourier transform versus a Laplace transform?  I am not a mathematician, so the little intuition I have tells me that it could be related to the boundary conditions imposed on the solution I am trying to find, but I am unable to state this rigorously or find a reference that discusses this.  Any help would be appreciated.  Thanks.

REPLY [6 votes]: Fourier transform (FT) - (roughly) a tool to visualize ANY signal as a sum of sinusoids.
Laplace transform (LT) - a tool to analyze the stability of systems.
Why can't we use FT to analyze systems? because it cannot handle exponentially growing signals. That in turn is because a signal needs to be absolutely integrable(necessary, but not sufficient condition) for it to have a FT.
On the other hand, LT is specially designed (by Laplace himself after he rejected the thesis of Fourier and extended FT) to handle exponentially growing signals so that now you could if a system has an exponentially growing output ( $a > 0$  where $s = a + jw$) or exponentially decaying output ($a < 0$).
Therefore if you have a square wave type signal and you wanted to break it up into sinusoids of various frequencies, you'd use FT and on the other hand, if you have a system and you want to understand how stable it is, then you'd try to get LT of it's impulse response to further analyze the systems relative stability.