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

2018 ◽  
Vol 12 ◽  
pp. 35-68
Author(s):  
Antonio Granata
Keyword(s):  
Asymptotic Expansions ◽  
Asymptotic Behaviors ◽  
The Real ◽  
Special Cases ◽  
Real Domain ◽  
Rapidly Varying Functions

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.

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

2017 ◽  
Vol 9 ◽  
pp. 1-33 ◽  
Author(s):  
Antonio Granata
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Expansions ◽  
Higher Order ◽  
Weight Functions ◽  
Asymptotic Behaviors ◽  
The Real ◽  
Special Cases ◽  
Real Domain ◽  
The Given ◽  
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.

Asymptotic expansions and converging factors V. Lommel, Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 284-292 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Lommel Functions ◽  
Special Cases

A theory of Lommel functions is developed, based upon the methods described in the first four papers (I to IV) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘basic converging factors’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions.

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