In this paper we study the problem of exponentially small splitting of separatrices of one degree
of freedom classical Hamiltonian systems with a non-autonomous perturbation which is fast and
periodic in time. We provide a result valid for general systems which are polynomials or trigonometric
polynomials in the state variables. Our result consists in obtaining a rigorous proof of the asymptotic
formula for the measure of the splitting. We have obtained that the splitting has the asymptotic
behavior K" e−a/" identifying the constants K, and a in terms of the features of the system.
The study of our problem leads us to consider several cases. In some cases, assuming the per-
turbation is small enough, it turns out that the values of K, coincides with the classical Melnikov
approach. We have identified the limit size of the perturbation for which this classical theory holds
true. However for the limit cases, which appear naturally both in averaging theory and bifurcation
theory, we encounter that, generically, neither K nor are actually well predicted by Melnikov theory.