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

DOI: 10.4236/apm.2015.52013   PDF   HTML   XML   5,069 Downloads   5,559 Views   Citations

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.

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Granata, A. (2015) The Role of Asymptotic Mean in the Geometric Theory of Asymptotic Expansions in the Real Domain. Advances in Pure Mathematics, 5, 100-119. doi: 10.4236/apm.2015.52013.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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