As an additional note, there are a few different
definitions of the
Laplace transform. The one that seems to appear in mathematics classes is the one listed above; the range of integration is from 0 to
infinity.
However, some sources, such as my book for the Electrical Engineering Signals and Systems course I took, define the Laplace transform to range from
negative infinity to infinity. This is referred to as the
bilateral transform, as opposed to the unilateral transform described above. The two have identical properties; which to use depends on one's application.
Initial value problems require the unilateral transform; it seems the bilateral transform is more useful for
signals and systems.
The
Fourier transform and the Laplace transform are closely related; in a sense, the Fourier transform can be seen as a
special case of the
bilateral Laplace transform, where the
complex variable s in the integral is restricted to be on the imaginary axis. Because of this, Laplace transforms apply to a larger set of functions than Fourier transforms, which can run into
discontinuities along the imaginary axis that cause either the transform or its
inverse to be
divergent.
In control theory, a system is stable only if all the
poles of the
system function, which is the Laplace transform of the system's
impulse response, are on the left side of the
imaginary axis. This is a basic tool of
Classical Control