A simple approach to the numerical Laplace transform inversion
The numerical inversion of the Laplace transform is a well-known ill-posed problem, due to an improper contour integral in the inversion formula. A number of methods were proposed in the 1960s and are still used, but few people have worked on the problem over the last few decades. We will first discuss the Weeks' method, which deforms the contour to the unit circle and computes a Laguerre expansion of the target function, and issues with its implementation. We will then discuss a naive approach of adaptive integration directly on some finite portion of the contour and compare results.