  In this paper we study the splitting of separatrices phenomenon which arises
when one considers a Hamiltonian System of one degree of freedom with a fast
periodic or quasiperiodic and meromorphic in the state variables perturbation.
The obtained results are different from the previous ones in the literature,
which mainly assume algebraic or trigonometric polynomial dependence on the
state variables. As a model, we consider the pendulum equation with several
meromorphic perturbations and we show the sensitivity of the size of the
splitting on the width of the analyticity strip of the perturbation with
respect to the state variables. We show that the size of the splitting is
exponentially small if the strip of analyticity is wide enough. Furthermore, we
see that the splitting grows as the width of the analyticity strip shrinks,
even becoming non-exponentially small for very narrow strips. Our results
prevent from using polynomial truncations of the meromorphic perturbation to
compute the size of the splitting of separatrices.
