  We present the first two leading terms of the 1/N (genus) expansion of the
free energy for ensembles of normal and complex random matrices. The results
are expressed through the support of eigenvalues (assumed to be a connected
domain in the complex plane). In particular, the subleading (genus-1) term is
given by the regularized determinant of the Laplace operator in the
complementary domain with the Dirichlet boundary conditions. An explicit
expression of the genus expansion through harmonic moments of the domain gives
some new representations of the mathematical objects related to the Dirichlet
boundary problem, conformal analysis and spectral geometry.
