# Using Complex Integration to Figure Inverse Laplace Transform

## Abstract

The process of figuring inverse Laplace transform is the counterpart of that of figuring the transform itself! In the latter we integrate the function against *e*^{−st} but over time. In the former we integrate the Laplace transform against the function *e*^{ st } but now over the complex frequency *s* = *σ* + *jω*. In carrying on the integration we must pick a *σ* and then integrate over the *jω* −axis. Most of the time the resulting integral is non-integrable by conventional means. To accomplish we first convert the linear integral to a contour one, ensure that the arc-part of the integral dies out, and then utilize and harness the Cauchy residue theorem. This theorem ties the contour integral (which is essentially the inverse transform) to the residues residing within the integral. Finding the residues is systematic and depends on the order of the poles. We demonstrate this whole process with a few examples and illustrate the particulars of the contour integration. We cover the cases of real and complex poles and for all cases compare resulting time signal to expected one.