Asymptotic expansion of the period function II

Let P be a not necessarily bounded polycycle of an analytic vector field on an open set of the plane. Suppose that the singularities which appear after desingularization of the vertices of P are hyperbolic. Consider the function T defined by the return time near P. It is shown that the function T and its derivative T′ have asymptotic expansions similar to the series of Dulac but with negative powers.