Laplace / Z-Transform: ROC, Poles & Stability

A pole-zero map is ambiguous: only the ROC — always a pole-free strip (Laplace) or ring (Z) whose side is fixed by causality or absolute-summability — pins down the signal and its stability verdict; two-sided signals intersect their causal and anti-causal ROCs, and a causal impulse response is read straight off the power-series (long-division) expansion of H(z).