Many texts (e.g. 1-2-3) on random matrices start with some variation of the identity:
$$\rho_1(\lambda) = \frac{1}{\pi} \text{Im}\{\langle\text{Tr}(\mathbf{X}-\lambda\mathbf{I})^{-1}\rangle\}$$
where the brakets denote the average over the ensemble of eigvenvalues and 'Tr' is the trace of the resolvant matrix $(\mathbf{X}-\lambda\mathbf{I})^{-1}$, where $\lambda\in\mathbb{C}\backslash\mathbb{R}$. The left side is the 1 point correlation function, i.e. the density, which is also defined as
$$\rho_1(\lambda)= \big\langle\sum_i \delta(\lambda-\lambda_i)\big\rangle $$
(same ensemble). How does one prove the above identity?
I provided a partial proof on Math Overflow, and Carlo Beenakker corrected some mistakes related to complex integration and clashing definitions of the marginal probability density. I consider this question answered.