  In this paper we consider general nearly integrable analytic Hamiltonian
systems of one and a half degrees of freedom which are a trigonometric
polynomial in the angular state variable. In the resonances of these systems
generically appear hyperbolic periodic orbits. We study the possible
transversal intersections of their invariant manifolds, which is exponentially
small, and we give an asymptotic formula for the measure of the splitting. We
see that its asymptotic first order is of the form $K \varepsilon^{\beta}
\text{e}^{-a/\varepsilon}$ and we identify the constants $K,\beta,a$ in terms
of the system features. We compare our results with the classical Melnikov
Theory and we show that, tipically, in the resonances of nearly integrable
systems Melnikov Theory fails to predict correctly the constants $K$ and
$\beta$ involved in the formula.

Dynamical Systems