I struggle to complete the last step of the derivation of Wigner's semi circle law (or the Marcenko-Pastur density for that matter), from the corresponding Stieltjes transform.
The Stieltjes transform of the semi circle law is given by
$\mathcal{S} = (z\pm\sqrt{z^{2}-4\sigma^{2}})/2\sigma^{2} $
I know that $\lim_{\eta \rightarrow 0}\mathcal{S}(x\pm i\eta) = \mathcal{h}(x) -\pm \rho(x)$
and that $\rho(x) \equiv (1/\pi) \lim_{\eta \to 0} \operatorname{Im}(\mathcal{S}(x-i\eta))$
I just don't see how to get from one to the other, I always end up in a horrendous algebraic mess and every reference
p.22 https://arxiv.org/pdf/1610.08104.pdf
https://terrytao.wordpress.com/2010/02/02/254a-notes-4-the-semi-circular-law/
seems to just say it is "easily" read off.
As is often the case, you see the answer only when you ask for help.
It really is as simple as applying the definition, obviously for z, it doesn't matter in the limit so you just take out a factor of i in the square root and divide through by pi. This is worked out in full on p.55 of https://arxiv.org/pdf/1712.07903.pdf