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
{\displaystyle S_{\rho }(z)=\int _{I}{\frac {\rho (t)\,dt}{t-z}},\qquad z\in \mathbb {C} \setminus I.}
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
lim
ε →
( x − i ε ) −
2 i π
If the measure of density ρ has moments of any order defined for each integer by the equality
{\displaystyle m_{n}=\int _{I}t^{n}\,\rho (t)\,dt,}
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:
The correspondence
{\textstyle (f,g)\mapsto \int _{I}f(t)g(t)\rho (t)\,dt}
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
{\displaystyle Q_{n}(x)=\int _{I}{\frac {P_{n}(t)-P_{n}(x)}{t-x}}\rho (t)\,dt.}
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.)