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 *x*_{0} ∈ 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.

Periodical:

International Journal of Advanced Research in Mathematics (Volume 9)

Pages:

1-33

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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Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License

References:

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