= Bromwich inversion formula
{c}
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,
$$
f(t)=\frac1{2\pi i}\lim_{R\to\infty}\int_{\gamma-iR}^{\gamma+iR}e^{st}F(s)\,ds.
$$
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.<nist-digital-library-of-mathematical-functions/1 Algebraic and Analytic Methods>[§1.14(iii), Laplace inversion and bounded-variation sufficient conditions]{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.
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