Whatever you said is correct.But the stability aspect is for a system.I quoted an example to show that σ is a variable and it has limits which is called REGION OF CONVERGENCE and the limits vary according to the function.For the system to be stable all roots of transfer function of respective system must belong to left half of splane.Consider the previous example .
LT{tu(t)}=1/s^2.The roots of this eq. are 0,0 which lies on jw axis hence this system,if designed is marginally stable.You may have ROC extending from some negative value to +∞ but the system is stable only when σ lies in left half of splane.
or you can understand like this. σ is only used to make equation integrable.After a transfer function is obtained, we should plot the poles of that T.F.Here,what you h've said is to be checked.Am I clear?
