I found the question below in a forum and I also have same problem. Unfortunately, there is no answer at all so I will post it here hope someone could make it clear.
A gain relation in a circuit of RCL and dependent sources ends up in an H(s) which is a quotient of polynomials in s. Number of poles is the number of energy storing elements independent of each other (you can assign independent starting conditions) and zeroes depend on the behavior of H(s) when s tends to infinity and the number of poles. Some zeroes and poles can be found by inspection, this is done knowing the above and observing some conditions and values of s so that the gain becomes zero or infinity.
s is supposed to be sigma + jw, and sigma arises so that the transform integral converges. However, when constructing bode plots, this is completely ignored. Why? Is there a physical meaning to sigma?
The effects on the bode plot of zeroes and poles are to change the slope in 20dB incrementals (bode magnitude), the gain isn't really infinite on the poles, since s is replaced by jw and if the poles are real, s being complex will never have those real values so that the bode plot goes infinite. What is the meaning of this?
A region of convergence (ROC) for s can be found, which is a range of values for s so that the integrals converge. What is the ROC useful for? What happens when the frequency is outisde the ROC? Textbooks are pretty good in showing how to find these ROCs, but not in making clear what effects they have in your circuit.