Stieltjes transform of a measure
= Stieltjes transform of a measure
{c}
{title2=$g_\mu(z)=\int_{\mathbb R}(x-z)^{-1}\,d\mu(x)$}
For a finite positive <measure> on the real line, this convention for its Stieltjes transform is analytic off the real line and has positive imaginary part in the upper half-plane if the measure is nonzero. Some sources use $(z-x)^{-1}$ instead, changing the sign. The transform of an <empirical spectral measure> equals the normalized trace of the <Stieltjes matrix resolvent>.