Abscissa of absolute convergence

Summary

The abscissa of absolute convergence $\sigma_a$ of a Laplace transform marks the left edge of the half-plane where $\int_0^\infty \abs{f(t)} e^{-\operatorname{Re}(s) t}\, dt$ is finite. It is never smaller than the abscissa of convergence $\sigma_c$. In the half-plane of absolute convergence the transform can be handled with the standard theorems of Lebesgue integration, such as dominated convergence and Fubini's theorem. The strip $\sigma_c < \operatorname{Re} s < \sigma_a$, where the integral converges only conditionally, can be nonempty.