The existence of Laplace transform of a given depends on whether
the transform integral converges

which in turn depends on the duration and magnitude of as well as the real part of (the imaginary part of determines the frequency of a sinusoid which is bounded and has no effect on the convergence of the integral).

**Right sided signals: **
may have infinite duration for
, and a positive tends to attenuate
as
.

**Left sided signals: **
may have infinite duration for
, and a negative tends to attenuate
as
.

Based on these observations, we can get the following properties for the ROC:

- If is absolutely integrable and of finite duration, then the ROC is
the entire s-plane (the Laplace transform integral is finite, i.e.,
exists, for any ).
- The ROC of consists of strips parallel to the -axis in the
s-plane.
- If is right sided and
is in the ROC, then any to
the right of (i.e.,
) is also in the ROC, i.e.,
ROC is a right sided half plane.
- If is left sided and
is in the ROC, then any to
the left of (i.e.,
) is also in the ROC, i.e.,
ROC is a left sided half plane.
- If is two-sided, then the ROC is the intersection of the two one-sided
ROCs corresponding to the two one-sided components of . This intersection
can be either a vertical strip or an empty set.
- If is rational, then its ROC does not contain any poles (by
definition
dose not exist). The ROC is bounded by
the poles or extends to infinity.
- If is a rational Laplace transform of a right sided function ,
then the ROC is the half plane to the right of the rightmost pole; if
is a rational Laplace transform of a left sided function ,
then the ROC is the half plane to the left of the leftmost pole.
- A signal is absolutely integrable, i.e., its Fourier transform
exists (first Dirichlet condition, assuming the other two
are satisfied), if and only if the ROC of the corresponding Laplace
transform contains the imaginary axis or .

**Example 1: ** Consider the Laplace transform of a two-sided signal
:

The Laplace transform of the two components can be obtained from the two examples discussed above. From example 1, we get

and let in example 2, we have

Combining the two components, we have

Whether exists or not depends on . If , i.e., decays exponentially as , then the ROC is the strip between and and exists. But if , i.e., grows exponentially as , then the ROC is an empty set and does not exist.

**Example 2: ** Given the following Laplace transform, find the corresponding
signal:

There are three possible ROCs determined by the two poles and :

- The half plane to the right of the rightmost pole , with the
corresponding right sided time function

- The half plane to the left of the leftmost pole , with the
corresponding left sided time function

- The vertical strip between the two poles
, with the
corresponding two sided time function