  Brezin-Hikami contour-integral representation of exponential multidensities
in finite N Hermitian matrix model is a remarkable implication of the old
Hermitian-Kontsevich duality. It is also a simplified version of Okounkov's
formulas for the same multidensities in the cubic Kontsevich model and of
Nekrasov calculus for LMNS integrals, a central piece of the modern studies of
AGT relations. In this paper we use Brezin-Hikami representation to derive
explicit expressions for the Harer-Zagier multidensities (from
arXiv:0906.0036): the only known exhaustive generating functions of all-genera
Gaussian correlators which are fully calculable and expressed in terms of
elementary functions. Using the Brezin-Hikami contour integrals, we rederive
the 1-point function of Harer and Zagier and the 2-point arctangent function of
arXiv:0906.0036. We also present (without a proof) the explicit expression for
the 3-point function in terms of arctangents. Derivation of the 3-point and
higher Harer-Zagier functions remains a challenging problem.
