An extension of the Henrici formula for Laplace transform inversion

Under certain conditions, starting from the Riemann inversion formula, which gives an explicit representation of the inverse Laplace transform in the complex form, we derive an integral equation, of convolution type, whose solution is the inverse Laplace transform function.

This formula can be used if the Laplace transform has a finite number of singularities, located everywhere in the complex plane, and provided that their corresponding residues are known. It only requires the knowledge of the Laplace transform function on the real negative axis. Preliminary numerical experiments illustrating the reliability of the inversion algorithm are described.