Let $0\le \lambda\le 1$ throughout.
The background is as follows. The Marchenko-Pastur (or free Poisson) law $\mathrm{d}\mu$ equals $(1-\lambda)\delta_0+\mathrm{d}\nu$, where $\mathrm{d}\nu=\frac{1}{2\pi t}\sqrt{4\lambda-(t-1-\lambda)^2}\ \mathrm{d}t$ with support $[(1-\sqrt{\lambda})^2,(1+\sqrt{\lambda})^2]$. In particular, $\mu(\{0\})=1-\lambda$. (see wikipedia page where $\alpha=1$ is taken.)
The Cauchy transform (i.e. the negative of the Stieltjes transformation) of this distribution is given by $$ G_{\mu}(z)=\frac{z+1-\lambda-\sqrt{(z-1-\lambda)^2-4\lambda}}{2z}. $$
The residue of $G_\mu(z)$ at $z=0$ should be $1-\lambda$; How to calculate this??? (My calculation shows the residue is 0.)