Strip of convergence of the bilateral transform

Remarks

The integral defining $\mathcal{B}\{f\}(s)$ converges absolutely for every $s \in S$, and $\mathcal{B}\{f\}$ is holomorphic on $S$. The integral may also converge on one or both boundary lines; these points are not counted in the open strip. Example: for $f(t) = e^{-\abs{t}}$, one finds $\alpha = -1$ and $\beta = 1$, so $S = \{-1 < \operatorname{Re} s < 1\}$. Non-example: for $f = 1$, $\alpha = 0$ and $\beta = 0$, so the strip is empty, and indeed the two-sided integral converges nowhere. The strip is part of the data of a transform. The function $1/(1 - s)$ on $\operatorname{Re} s < 1$ inverts to $e^t$ on $t < 0$ and $0$ on $t > 0$, while on $\operatorname{Re} s > 1$ it inverts to $-e^t$ on $t > 0$ and $0$ on $t < 0$.