The Generalized Stieltjes Transform and Its Inverse

TL;DR

This paper generalizes the inverse of the generalized Stieltjes transform for all positive parameters, providing a simpler one-dimensional integral form that extends previous specific cases.

Contribution

It derives a unified, simplified inverse transform formula for the GST applicable to all positive  ho, expanding on prior work limited to  ho=3/2.

Findings

Unified inverse formula for all  ho > 0

Simpler one-dimensional integral representation

Extension of previous specific inverse transform results

Abstract

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.