Exponential asymptotics

Recently, there has been a surge of practical and theoretical interest on the part of mathematical physicists, classical analysts and abstract analysts in the subject of exponential asymptotics, or hyperasymptotics, by which is meant asymptotic approximations in which the error terms are relatively exponentially small. Such approximations generally yield much greater accuracy than classical asymptotic expansions of Poincare type, for which the error terms are algebraically small: in other words, they lead to "exponential improvement." They also enjoy greater regions of validity and yield a deeper understanding of other aspects of asymptotic analysis, including the Stokes phenomenon. We shall obtain readily-applicable theories of hyperasymptotic expansions of solutions of differential equations and for integrals with one or more saddles. The main tool will be the Borel transform, which transforms the divergent asymptotic expansions into convergent series. Other methods will also be mentioned. Topics to be discussed are: Least terms in divergent asymptotic expansions; Exponentially-improved asymptotic expansions; Smoothing of the Stokes phenomenon; Resurgence; Computation of Stokes multipliers (connection coefficients).