Abscissa of convergence

Remarks

If the Laplace integral converges at a complex number $s_0$, it converges at every $s$ with $\operatorname{Re} s > \operatorname{Re} s_0$. Hence the integral converges for $\operatorname{Re} s > \sigma_c$ and diverges for $\operatorname{Re} s < \sigma_c$. Example: for $f(t) = e^{at}$ with $a \in \R$, $\int_0^T e^{(a - \sigma)t}\, dt$ has a finite limit exactly when $\sigma > a$, so $\sigma_c = a$. Non-example of a finite value: for $f(t) = e^{t^2}$ the integral diverges for every real $\sigma$, so $\sigma_c = +\infty$; for $f(t) = e^{-t^2}$ it converges for every real $\sigma$, so $\sigma_c = -\infty$. For $f = 1$, $\sigma_c = 0$, and on the line $\operatorname{Re} s = 0$ the integral diverges at every point.