Complements to the Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions

Abstract

The purpose of this paper is to add some complements to the general theory of higher-order types of asymptotic variation developed in two previous papers so as to complete our elementary (but not too much!) theory in view of applications to the theory of finite asymptotic expansions in the real domain, the asymptotic study of ordinary differential equations and the like. The main results concern: 1) a detailed study of the types of asymptotic variation of an infinite series so extending the results known for the sole power series; 2) the type of asymptotic variation of a Wronskian completing the many already-published results on the asymptotic behaviors of Wronskians; 3) a comparison between the two main standard approaches to the concept of "type of asymptotic variation": via an asymptotic differential equation or an asymptotic functional equation; 4) a discussion about the simple concept of logarithmic variation making explicit and completing the results which, in the literature, are hidden in a quite-complicated general theory.

Operations with Higher-Order Types of Asymptotic Variation: Filling Some Gaps

“…In three previous papers [1] [2] [3] we developed a general theory of higher-order types of asymptotic variation for functions differentiable a certain number of times on some interval [ ) , T +∞ , spending much effort on results about operations with such classes of functions in [2] and pointing out some elementary applications of the general theory to integrals and sums. Subsequently, in [4] [5], we applied the theory to the difficult problem of evaluating the exact asymptotic behaviors of Wronskians whose entries belong to one or more of the classes of regularly-, smoothly-or rapidly-varying functions.…”

“…For the reader's convenience we report the general notations and essential facts of the theory, already listed in [ [3] ( ) ( ) ( )…”