ABSTRACT
This chapter introduces expansions in ɛ-shadow of numbers by Francine Diener as a nonstandard alternative to asymptotic expansions of functions. It deals with the domain of validity of asymptotic approximations. As a consequence the matching property may be expressed by strong convergence theorems: convergence to an external number implies strong convergence. The property of strong convergence and matching may also be formulated for functions, and as such it is particularly relevant to solutions of singularly perturbed differential equations, which typically exhibit “slow” and “fast” behaviour. The chapter also deals with a matching theorem for sequences and explains a particular flexible singular perturbation for canard solutions, which are a particular kind of solution losing stability, while remaining bounded. The strong convergence theorems enable more general matching theorems.
