Asymptotic Behaviors of Wronskians and Finite Asymptotic Expansionsin the Real Domain - Part II: Mixed Scales and Exceptional Cases

In this second Part of our work we study the asymptotic behaviors of Wronskians involving both regularly- and rapidly-varying functions, Wronskians of slowly-varying functions and other special cases. The results are then applied to the theory of asymptotic expansions in the real domain.

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

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.

Integral properties of rapidly and regularly varying functions

Regularly and rapidly varying functions are studied as well as the asymptotic properties related to, several classical inequalities and integral sums.

The weak and the strong equivalence relation and the asymptotic inversion

In this paper we discuss the relationship between the weak and the strong asymptotic equivalence relation and the asymptotic inversion, for positive and measurable functions defined on a half-axis [a,+?) (a > 0). As the main results, we prove a certain characterizations of the functional class of all rapidly varying functions, as well as some other functional classes.