ABSTRACT
This chapter deals with inversion of the Laplace and Hankel transforms and their associated contour integration. The Laplace transform is a powerful method for solving a diffusion-like differential equation that contains a first order differential in time. The Hankel transform is typically used in problems that can be described in terms of cylindrical coordinates. Convolution in the time domain is equivalent to multiplication in the Laplace transform domain. The Taylor series centers around an analytic point with its circle of convergence extending to the nearest singular point; while the Laurent series centers around a singular point and is analytic within an annulus between two circles: the inner circle encloses the singular point and the outer circle extends to the next singular point. There are two ways to obtain the inverse Laplace transform. The first one is straightforward while the second one uses a branch-cut integration.
