  We propose an asymptotic expansion formula for matrix integrals, including
oscillatory terms (derivatives of theta-functions) to all orders. This formula
is heuristically derived from the analogy between matrix integrals, and formal
matrix models (combinatorics of discrete surfaces), after summing over filling
fractions. The whole oscillatory series can also be resummed into a single
theta function. We also remark that the coefficients of the theta derivatives,
are the same as those which appear in holomorphic anomaly equations in string
theory, i.e. they are related to degeneracies of Riemann surfaces. Moreover,
the expansion presented here, happens to be independent of the choice of a
background filling fraction.
