Talbot's Method

Despite being widely accepted as a highly accurate method, in the twenty-five years since its introduction, the Talbot method has received relatively little attention WEIDEMAN1999. Unlike the Fourier series or Post-Widder approaches, Talbot's method is based on a deformation of the Bromwich contour. The idea is to replace the contour with one which opens toward the negative real axis

where
and .
It also requires that

It thus inherently assumes that the physical system damps highly oscillatory terms and is not appropriate for purely conservative problems. From a numerical perspective however, damping the highly oscillatory terms in the inversion integral is a stabilizing procedure since it is these which provide the largest contribution to the error when one performs numerical quadrature of the Laplace inversion integral.

The primary difficulty of the Talbot method is the selection of accurate values for the contour parameters. The original paper by Talbot is rather lengthly with only a few pages on the derivation of the contour using the steepest-decent method. The remaining pages are a detailed discussion of how to determine the parameters from the properties of the function to invert. The nontrivial task of developing software to select optimal values for these parameters has been undertaken for scalar functions MURLI1990 but the determination of these coefficients relies heavily on the known location of the singularities of the Laplace space functions. In response, many attempts have been made to simply Talbot's method or provide alternative integration contours EVANS2000,VALKO2003.