  Borel summation techniques are developed to obtain exact invariants from
formal adiabatic invariants (given as divergent series in a small parameter)
for a class of differential equations, under assumptions of analyticity of the
coefficients; the method relies on the study of associated partial differential
equations in the complex plane. The type and location of the singularities of
these associated functions, important in determining exponentially small
corrections to formal invariants are also briefly discussed.
