Analytic theory of finite asymptotic expansions in the real domain. Part I: two-term expansions of differentiable functions

Abstract

It is our aim to establish a general analytic theory of asymptotic expansions of type f(x)=a_1 phi_1(x)+dots+ a_n phi_n(x)+o(phi_n(x)), x tends to x_0 (*), where the given ordered n-tuple of real-valued functions phi_1 dots,phi_n forms an asymptotic scale at x_0. By analytic theory, as opposed to the set of algebraic rules for manipulating finite asymptotic expansions, we mean sufficient and/or necessary conditions of general practical usefulness in order that (*) hold true. Our theory is concerned with functions which are differentiable (n-1) or n times and the presented conditions involve integro-differential operators acting on f, phi_1, dots, phi_n. We essentially use two approaches; one of them is based on canonical factorizations of nth-order disconjugate differential operators and gives conditions expressed as convergence of certain improper integrals, very useful for applications. The other approach, valid for (n-1)-time differentiable functions starts from simple geometric considerations (as old as Newton's concept of limit tangent) and gives conditions expressed as the existence of finite limits, as x tends to x_0, of certain Wronskian determinants constructed with f, phi_1, dots, phi_n. There is a link between the two approaches and it turns out that the integral conditions found via the factorizational approach have striking geometric meanings. Our theory extends to general expansions the theory of polynomial asymptotic expansions thoroughly investigated in a previous paper. In the first part of our work we study the case of two comparison functions phi_1, phi_2. The theoretical background for the two-term theory is much simpler than that for n >=3 and, in addition, it is unavoidable to separate the treatments as the two-term formulas must be explicitly written lest they become unreadable.