In mathematics, the Stieltjes transformation of a measure of density on a real interval is the function of the complex variable defined outside by the formula

Inverse formula

Under certain conditions we can reconstitute the density function starting from its Stieltjes transformation thanks to the inverse formula of Stieltjes–Perron. For example, if the density is continuous throughout

Derivation of formula

is the probability density function of a distribution—a Cauchy distribution, these tend to a Dirac distribution against that is also known as the Poisson kernel (for the half-plane).

is absolutely continuous.

Connections with moments of measures

If the measure of density has moments

then the Stieltjes transformation of admits for each integer the asymptotic

Under certain conditions the complete expansion as a Laurent series

Relationships to orthogonal polynomials

defines an inner product on the space of continuous functions on the interval .

If is a sequence of orthogonal polynomials for this product, we can create the sequence of associated secondary polynomials

is a Padé approximation of

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 .

The Stieltjes transformation can also be used to construct from the density an effective measure for transforming the secondary polynomials into an orthogonal system. (For more details see the article secondary measure.)

See also

• Orthogonal polynomials

• Secondary polynomials

• Secondary measure

• Hilbert transform

References