Bromwich integral

Summary

The Bromwich integral is the complex line integral $\frac{1}{2 \pi i} \int_{\gamma - i \infty}^{\gamma + i \infty} e^{s t} F(s) \, ds$ along a vertical line $\operatorname{Re} s = \gamma$ in the half-plane where a Laplace transform $F$ is defined. It is the standard integral formula for the inverse Laplace transform: under mild conditions it returns the function $f$ with $\mathcal{L}\{f\} = F$. It also opens the way to computing inverse transforms by contour integration and residues.