Let g : R ¿ C be a C8-function with all derivatives bounded and let trn denote the normalized trace on the n × n matrices. In Ref. 3 Ercolani and McLaughlin established asymptotic expansions of the mean value ¿{trn(g(Xn))} for a rather general class of random matrices Xn, including the Gaussian Unitary Ensemble (GUE). Using an analytical approach, we provide in the present paper an alternative proof of this asymptotic expansion in the GUE case. Specifically we derive for a random matrix Xn that

where k is an arbitrary positive integer. Considered as mappings of g, we determine the coefficients aj(g), j ¿ N, as distributions (in the sense of L. Schwarts). We derive a similar asymptotic expansion for the covariance Cov{Trn[f(Xn)], Trn[g(Xn)]}, where f is a function of the same kind as g, and Trn = n trn. Special focus is drawn to the case where and for ¿, µ in C\R. In this case the mean and covariance considered above correspond to, respectively, the one- and two-dimensional Cauchy (or Stieltjes) transform of the .