85 Lerch’s Uniqueness Theorem

Suppose that $f$ and $g$ satisfy $(E)$ and that there exists $s_0$ such that

Then $f(x)=g(x)$ for all $x>0$.

There is an inversion formula, credited to Bromwich, which is rather hard to use. So if one knows that $F(\lambda )$ occurs as $ℒ(f)(\lambda )$ for some $f$, then the best way to find $f$ is by comparing $F$ with known Laplace transforms in tables.