In mathematics, the Stieltjes transformation Sρ(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

Connections with moments of measures

If the measure of density ρ has moments of any order defined for each integer by the equality then the Stieltjes transformation of ρ admits for each integer n the asymptotic expansion in the neighbourhood of infinity given by Under certain conditions the complete expansion as a Laurent series can be obtained:

Relationships to orthogonal polynomials

The correspondence defines an inner product on the space of continuous functions on the interval I. If {Pn} is a sequence of orthogonal polynomials for this product, we can create the sequence of associated secondary polynomials by the formula It appears that is a Padé approximation of Sρ(z) in a neighbourhood of infinity, in the sense that Since these two sequences of polynomials satisfy the same recurrence relation in three terms, we can develop a continued fraction for the Stieltjes transformation whose successive convergents are the fractions Fn(z) . The Stieltjes transformation can also be used to construct from the density ρ an effective measure for transforming the secondary polynomials into an orthogonal system. (For more details see the article secondary measure.)