region of convergence

When you write down the defining integral of a Laplace transform, integral from 0 to infinity of e^{-s t} f(t) dt, it does not automatically make sense for every s. The region of convergence is the set of complex numbers s for which that integral actually converges — the territory where the transform is genuinely defined by its integral rather than by later extension.

For the one-sided Laplace transform this region is always a right half-plane: there is a real number sigma_0, the abscissa of convergence, such that the integral converges whenever the real part of s exceeds sigma_0. The reason is the weight e^{-s t}: its size is governed by e^{-(Re s) t}, so making Re s large enough forces decay fast enough to tame f(t). For e^{a t} the threshold is Re s > a; for a function bounded by a constant, Re s > 0 suffices. A function that grows faster than any exponential (like e^{t^2}) has no abscissa at all and simply has no Laplace transform.

The region of convergence matters because it is the honest fine print on every transform pair. Two different time functions can share the same algebraic formula F(s) but be distinguished by their regions, and the inverse-transform Bromwich contour must be placed inside the region of convergence to recover the right f(t). In practice, once you know the transform converges somewhere, the function F(s) is extended to the rest of the s-plane by analytic continuation, which is why the tabulated formulas look like they hold everywhere even though the original integral does not.