TITLE:
The Role of Asymptotic Mean in the Geometric Theory of Asymptotic Expansions in the Real Domain
AUTHORS:
Antonio Granata
KEYWORDS:
Asymptotic Expansions, Formal Differentiation of Asymptotic Expansions, Regularly-Varying and Rapidly-Varying Functions, Asymptotic Mean
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.5 No.2,
February
26,
2015
ABSTRACT:
We
call “asymptotic mean” (at +∞) of a real-valued function the number, supposed to exist, ,
and highlight its role in the geometric theory of asymptotic expansions in the
real domain of type (*) where the comparison functions ,
forming an asymptotic scale at +∞, belong to one of the three classes having a definite “type of variation” at +∞, slow, regular
or rapid. For regularly varying comparison functions we can characterize
the existence of an asymptotic expansion (*) by the nice property that a
certain quantity F（t) has an asymptotic mean at +∞. This quantity is
defined via a linear differential operator in f and admits of a remarkable geometric interpretation as it
measures the ordinate of the point wherein that special curve ,
which has a contact of order n - 1
with the graph of f at the generic
point t, intersects a fixed vertical
line, say x = T. Sufficient or necessary conditions hold true for the other two
classes. In this article we give results for two types of expansions already
studied in our current development of a general theory of asymptotic expansions
in the real domain, namely polynomial and two-term expansions.