The inverse Laplace transform is given by the Bromwich Integral:

where is a real constant chosen to ensure convergence of the integral.

Solving Inverse Laplace Transform by Pattern Matching

Solving the Bromwich integral requires integration in complex plane, which can be complicated. Though we can often evaluating inverse Laplace transform by pattern-matching with the table of inverse Laplace transform.

Inverse Laplace Transforms and ROCs

Given an function in the s-plane, the inverse Laplace transform of the unilateral Laplace transform is unique. This uniqueness is due to the implicit assumption of causality in the unilateral transform, which restricts the region of convergence to a right-half plane in the s-plane.

By contrast, for a bilateral Laplace transform, the inverse Laplace transform is not unique and depend on the region of convergence.