Stieltjes transform

Summary

The Stieltjes transform of a finite measure $\mu$ on the real line is the function $S_\mu(z) = \int \frac{d\mu(t)}{t - z}$ of a complex variable $z$ off the real axis. It is a holomorphic function that encodes $\mu$ completely, and the measure can be read back from its boundary values on the real axis. It is central in the moment problem, in spectral theory, and in random matrix theory, where limiting eigenvalue distributions are found through their Stieltjes transforms. For a function $f$ on $(0, \infty)$ the classical form is $\int_0^\infty \frac{f(t)}{x + t}\, dt$, which is the Laplace transform applied twice.