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

DOI: 10.4236/apm.2015.52013, PP. 100-119

Keywords: Asymptotic Expansions, Formal Differentiation of Asymptotic Expansions, Regularly-Varying and Rapidly-Varying Functions, Asymptotic Mean

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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 -

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