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.
References
[1]
Granata, A. (2016) The Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions. Part I: Higher-Order Regular, Smooth and Rapid Variation. Advances in Pure Mathematics, 6, 776-816. https://doi.org/10.4236/apm.2016.612063
[2]
Granata, A. (2016) The Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions. Part II: Algebraic Operations and Types of Exponential Variation. Advances in Pure Mathematics, 6, 817-867. https://doi.org/10.4236/apm.2016.612064
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Granata, A. (2018) Asymptotic Behaviors of Wronskians and Finite Asymptotic Expansions in the Real Domain. Part II: Mixed Scales and Exceptional Cases. International Journal of Advanced Research in Mathematics, 12, 35-68. https://doi.org/10.18052/www.scipress.com/IJARM.12.35
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