Adiabatic invariant of the harmonic oscillator, complex matching and resurgence
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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.