Stieltjes Transformation

In mathematics, the Stieltjes transformation S_ρ(z) of a measure of density ρ on a real interval I is the function of the complex variable z defined outside I by the formula

Under certain conditions we can reconstitute the density function ρ starting from its Stieltjes transformation thanks to the inverse formula of Stieltjes-Perron. For example, if the density ρ is continuous throughout I, one will have inside this interval