where is real and
is a complex variable. The
one-sided Laplace transform is also called the unilateral
Laplace transform. There is also a two-sided, or
bilateral, Laplace transform obtained by setting the lower
integration limit to instead of 0. Since we will be
analyzing only causal^{B.1} linear systems using the Laplace transform, we can use
either. However, it is customary in engineering treatments to use the
one-sided definition.

When evaluated along the axis (i.e., ), the
Laplace transform reduces to the unilateral Fourier transform:

The Fourier transform is normally defined bilaterally (
above), but for causal signals , there is no
difference. We see that the Laplace transform can be viewed as a
generalization of the Fourier transform from the real line (a simple
frequency axis) to the entire complex plane. We say that the Fourier
transform is obtained by evaluating the Laplace transform along the
axis in the complex plane.

An advantage of the Laplace transform is the ability to transform signals which
have no Fourier transform. To see this, we can write the Laplace
transform as

Thus, the Laplace transform can be seen as the Fourier transform of an
exponentially windowed input signal.
For (the so-called ``strict right-half plane'' (RHP)), this
exponential weighting forces the Fourier-transformed signal toward
zero as
. As long as the signal does not increase
faster than for some , its Laplace transform will exist for all
. We make this more precise in the next section.