Stieltjes inversion formula

Remarks

For $\varepsilon > 0$ one has $\frac{1}{\pi} \operatorname{Im} S_\mu(x + i\varepsilon) = \int_\R P_\varepsilon(x - t)\, d\mu(t)$ with the Poisson kernel $P_\varepsilon(u) = \frac{1}{\pi} \frac{\varepsilon}{u^2 + \varepsilon^2}$ of the upper half-plane. By Fubini, the left side before the limit is $\int_\R \frac{1}{\pi}\big(\arctan\frac{b - t}{\varepsilon} - \arctan\frac{a - t}{\varepsilon}\big)\, d\mu(t)$. The integrand is bounded by $1$ and tends to $1$ for $a < t < b$, to $\frac12$ for $t = a$ or $t = b$, and to $0$ otherwise, so dominated convergence gives the formula. If $\mu$ has a continuous density $\rho$ near $x$, then $\rho(x) = \lim_{\varepsilon \to 0^+} \frac{1}{\pi} \operatorname{Im} S_\mu(x + i\varepsilon)$.

Example: for $\mu = \delta_0$, $\frac{1}{\pi} \operatorname{Im} S_\mu(x + i\varepsilon) = \frac{1}{\pi} \frac{\varepsilon}{x^2 + \varepsilon^2}$, which concentrates at $0$ with total mass $1$.

Going deeper

For a student

The formula is the reason why the Stieltjes transform is the standard tool to identify a measure. In random matrix theory one computes the limit of the Stieltjes transforms of empirical eigenvalue distributions, and the inversion formula then gives the limiting law, such as the semicircle law. In the moment problem and in the spectral theory of self-adjoint operators the same formula recovers a spectral measure from its resolvent.