TITLE: Adiabatic Invariant of the Harmonic Oscillator, Complex Matching
and Resurgence.

AUTHORS: C. Bonet (1), D. Sauzin (2), T. Seara (1) and M. Val\`encia (1)

(1) Departament de Matem\`atica Aplicada I - UPC. Diagonal 647,
    08028-Barcelona. Spain.
(2) Bureau des Longitudes, CNRS, Paris. France.

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 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.