Laplace Transforms
In a nutshell Laplace transform is a generalization of Fourier transform. Laplace transforms can be applied to the analysis of many unstable systems and play an important role in the investigation of the stability or instability of systems.
The Laplace transform of a general signal x(t) is defined as
$ X(s) = \int_{-\infty}^{\infty}x(t){e^{-st}}\, dt $
Now that I have given a lose definition to what Laplace transforms are let me talk about some of the properties of the Region of Convergence for Laplace transforms.
Region of Convergence
The range of values of s for which the integral in the equation above converges is referred to as the region of convergence.
Properties
Property 1: The ROC X(s) consists of strips parallel to the jw-axis in the s-plane.
Property 2: For rational Laplace transforms, the ROC does not contain any poles
Property 3: If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s-plane.
Property 4: If x(t) is right sided, and if the line Re{s}=sigma(0) is in the ROC, then all values of s for which Re{s}>sigma(0) will also be in the ROC.
Property 5: If the Laplace transform X(s) of x(t) is rational, then its ROC is bounded by poles or extends to infinity. In addition, no poles of X(s) are contained in the ROC.
Property 6:If the Laplace transform X(S) of x(t) is rational, then if x(t) is right sided,the ROC is the region in the s-plane to the right of the rightmost pole. If x(t) is left sided, the ROC is the region in the s-plane to the left of the leftmost pole.
