Integral formulae for the eigenvalue density of complex random matrices

We show that the density of complex eigenvalues for unitary invariant ensembles of complex matrices A can be written as an integral over the eigenvalues gj of AA†. For the standard random matrix ensembles with matrix density of the form Πjw(gj), this integral can be further reduced to a twofold integral involving the Christoffel-Darboux kernel for the orthogonal polynomials associated with weight w. (fast track communication)