 Bromwich inversion formula

Let $f$ be of exponential order $a$, piecewise continuously differentiable on finite intervals of nonnegative time, and let $F$ be its unilateral Laplace transform. For $\gamma >a$ and $t>0$ at a point of continuity,At a jump the symmetric integral recovers $[f(t-)+f(t+)]/2$. The vertical line must lie in the defining region of absolute convergence of a Laplace transform; saying only that it avoids singularities of an analytic continuation does not supply that condition.^[ref]

To see the inversion mechanism, extend $e^{-\gamma t}f(t)$ by zero to negative time. Its Fourier transform at angular frequency $\omega$ is $F(\gamma +i\omega )$. The Fourier inversion theorem, followed by multiplication by $e^{\gamma t}$ and the substitution $s=\gamma +i\omega$, gives the displayed formula. Contour deformation is a subsequent step that needs separate growth estimates.