Analytical and numerical inversion of the Laplace-Carson transform by a differential method

A differential method is presented for recovering a function from its Laplace–Carson transform pf(p) given as continuous or discrete data on a finite interval. The introduction of the variable u=1/p converts this transform into a Mellin convolution, with a transformed kernel involving the gamma function G. The truncation of the infinite product representation of 1/G leads to an approximate differential expression for the solution. The algorithm is applied to selected analytical and numerical test problems; discrete and noisy data are differentiated with the aid of Tikhonov's regularization. For the inversion of a Laplace transform, the present formula is proved to be equivalent to the Post– Widder expression.