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Asymptotic Behaviors of Wronskians and Finite Asymptotic Expansions in the Real Domain - Part I: Scales of Regularly- or Rapidly-Varying Functions

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Abstract:

In a previous series of papers we established a general theory of finite asymptotic expansions in the real domain for functions f of one real variable sufficiently-regular on a deleted neighborhood of a point x0 ∈ R, a theory based on the use of a uniquely-determined linear differential operator L associated to the given asymptotic scale and wherein various sets of asymptotic expansions are characterized by the convergence of improper integrals involving both the operator L applied to f and certain weight functions constructed by means of Wronskians of the given scale. Very special cases apart, Wronskians have quite complicated expressions and unrecognizable asymptotic behaviors; however in the present work, split in two parts, we highlight some approaches for determining the exact asymptotic behavior of a Wronskian when the involved functions are regularly- or rapidly-varying functions of higher order. This first part contains: (i) some preliminary results on the asymptotic behavior of a determinant whose entries are asymptotically equivalent to the entries of a Vandermonde determinant; (ii) the fundamental results about the asymptotic behaviors of Wronskians involving scales of functions all of which are either regularly (or, more generally, smoothly) varying or rapidly varying of a suitable higher order. A lot of examples and applications to the theory of asymptotic expansions in the real domain are given.

Info:

Periodical:
International Journal of Advanced Research in Mathematics (Volume 9)
Pages:
1-33
DOI:
10.18052/www.scipress.com/IJARM.9.1
Citation:
A. Granata "Asymptotic Behaviors of Wronskians and Finite Asymptotic Expansions in the Real Domain - Part I: Scales of Regularly- or Rapidly-Varying Functions", International Journal of Advanced Research in Mathematics, Vol. 9, pp. 1-33, 2017
Online since:
Jun 2017
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