ABSTRACT

As we have seen, there is an important distinction between asymptotic expansions and asymptotic series. The operator f 7→ Ap(f) which associates to f its asymptotic power series is linear as seen in §1.1c. But it has a nontrivial kernel (Ap(f) = 0 for many nonzero functions), and thus the description through asymptotic power series is fundamentally incomplete. There is no unambiguous way to determine a function from its classical asymptotic series alone. On the other hand, the operator f 7→ A(f) which associates to f its asymptotic expansion has zero kernel, but it is still false that A(f) = A(g) implies f = g (A is not linear; see Remark 1.25). The description of a function through its asymptotic expansion is also incomplete.