Bromwich integral

The contour integral $\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}F(s)e^{st}\,ds$ that recovers $f(t)$ from its Laplace transform $F(s)$ .

The Bromwich integral is $\dfrac{1}{2\pi i}\displaystyle\int_{\gamma-i\infty}^{\gamma+i\infty}F(s)e^{st}\,ds$ , an integral straight up a vertical line $\operatorname{Re}(s)=\gamma$ in the complex plane, chosen with every pole of $F$ to its left. It is the exact inverse of the Laplace transform: closing this vertical line into a loop with a large left-opening arc and applying the residue theorem turns evaluating it into summing residues, the whole content of the Bromwich inversion formula.

Examples

For $F(s)=1/(s+2)$ , the Bromwich integral (with any $\gamma>-2$ ) evaluates to $e^{-2t}$ , matching the ordinary Laplace transform table entry -- now derived by contour integration instead of looked up.
Choosing a different valid $\gamma$ (any value with every pole of $F$ still to its left) never changes the Bromwich integral's value, since the region between two such vertical lines contains no pole to pick up an extra residue.