The generalized Stieltjes transform and its inverse

The generalized Stieltjes transform (GST) is an integral transform that depends on a parameter rho> 0. In previous work a convenient form of the inverse transformation was derived for the case rho= 3/2. This paper generalizes that result to all rho> 0. It is a well-known fact that the GST can be formulated as an iterated Laplace transform, and that therefore its inverse can be expressed as an iterated inverse Laplace transform. The form of the inverse transform derived here is a one-dimensional integral that is considerably simpler.