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The Role of Asymptotic Mean in the Geometric Theory of Asymptotic Expansions in the Real Domain ()

^{*}

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.

Keywords

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*Advances in Pure Mathematics*,

**5**, 100-119. doi: 10.4236/apm.2015.52013.

Conflicts of Interest

The authors declare no conflicts of interest.

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