Abscissa of convergence

Summary

The abscissa of convergence of a Laplace transform is a number $\sigma_c$ in $[-\infty, +\infty]$. The Laplace integral converges on the half-plane $\operatorname{Re} s > \sigma_c$ and diverges on the half-plane $\operatorname{Re} s < \sigma_c$. It plays the role that the radius of convergence plays for a power series. The same notion exists for Dirichlet series, which are Laplace–Stieltjes transforms of step functions. On the boundary line $\operatorname{Re} s = \sigma_c$ the integral may converge at some points and diverge at others.