TITLE:
Complements to the Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions
AUTHORS:
Antonio Granata
KEYWORDS:
Higher-Order Regularly-Varying Functions, Higher-Order Rapidly-Varying Functions, Smoothly-Varying Functions, Exponentially-Varying Functions, Logarithmically-Varying Functions, Asymptotic Differential Equations, As-ymptotic Functional Equations, Asymptotic Variation of Wronskians
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.9 No.5,
May
29,
2019
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.