C. Bonet, D. Sauzin, T.M. Seara, M. Valencia.
Adiabatic invariant of the Harmonic Oscillator,
Complex Matching and Resurgence.
(90K, LaTeX2e)

ABSTRACT.  The linear oscillator equation with a frequency
depending slowly on time is used to test
a method to compute exponentially small quantities. This work
present the matching method in the complex plane as a tool to obtain
rigorously the asymptotic variation of the action of the
associated hamiltonian {\sl beyond all orders}.
The solution in the complex plane is aproximated by a series in which all
terms present a singularity at the same point.
Following matching techniques near this singularity
one is led to an equation which does not depend on any
parameter, the so-called inner equation, of a Riccati type.
This equation is studied by resurgence methods.