, where the given ordered n-tuple of real-valued functions (ϕ1, ..., ϕn) forms an asymptotic scale at x0 ∈ . 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, ϕ1, ..., ϕn. 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 starts from simple geometric considerations and gives conditions expressed
as the existence of finite limits, as x → x0, of certain Wronskian determinants constructed with f, ϕ1, ..., ϕn. There is a link between the two approaches and it turns out that some of the integral conditions found via the factorizational
approach have geometric meanings. Our theory extends to more general expansions the theory of real-power asymptotic expansions
thoroughly investigated in previous papers. In the first part of our work we study the case of two comparison functions ϕ1, ϕ2 because the pertinent theory requires a very limited theoretical background and completely parallels the theory of polynomial
give either unsatisfactory or partial results when applied to an asymptotic expansion with at least two meaningful terms.
Simple examples show that some of these results are the best possible in the classical context. Hence a change of viewpoint
is necessary to arrive at useful results.
multiplication parameter, which by itself well indicates a certain extent of inherent inexactness of such a kind of kinetic evaluation. A better insight of this inquisitiveness was provided by the use of an asymptoticexpansion [ 88 ] of a series with a