Region of Convergence and Inverse Laplace

[Dustinsfl]

How do I find all the possible ROC for a transfer function written as
\[
H(s) = \frac{(s - 2)^{n_1}}{(s + 2)^{n_2}(s + 1)^{n_3}(s - 1)^{n_4}}
\]
where \(n_i\in\mathbb{N}\).

 

[Aaron Bex]

To find all the possible ROC of a transfer function written as above, you need to consider the cases where each \(n_i\in\mathbb{N}\) can take on different values. For example, if \(n_1 = 1, n_2 = 2, n_3 = 3,\) and \(n_4 = 4\), then the transfer function would be written as
\[
H(s) = \frac{(s - 2)^{1}}{(s + 2)^{2}(s + 1)^{3}(s - 1)^{4}}
\]
In this case, the ROC is given by the region of the complex plane in which the denominator has no poles (i.e., the zeros of the denominator). This region is given by
\[
|s + 2| > 0, |s + 1| > 0, |s - 1| > 0
\]
which means that the ROC is the region in the complex plane outside of the circles centered at \(-2\), \(-1\) and \(1\) with radii equal to one.

For other values of \(n_i\), the ROC will be similarly determined by the zeros of the denominator.