How to use Fourier transform in asymptotic analysis

This introductory paper presents a method for the analysis of differential equations with polynomial coefficients which also provides a funher insight into the Stokes Phenomenon. The method consists of a chain of steps based on the concept of the Stokes Structure and Fourier-like transforms adjusted to this Stokes Structure. Although the main object here is Bessel's equation our approach can be extended to more general matrix equations. It will be shown (i) how to derive the Stokes Structure directly from differential equations without any previous knowledge of Bessel or hypergeometric functions, (ii) how to adjust Fourier transforms to the Stokes Structure, (iii) how to answer questions on the interrelation between formal and actual solutions of Bessel's equation using Fourier Analysis, and finally (iv) how to evaluate the coefficient of the Stoke's Structure, thus providing a new insight into the Stokes Phenomenon

The Stokes structure in asymptotic analysis 1: Bessel, Weber and hypergeometric functions

This is the first in a series of four papers entitled 'The Stokes structure in asymptotic analysis'. They introduce a methodology for the asymptotic analysis of differential equations with polynomial coefficients which also provides a further insight into the Stokes' Phenomenon. This approach consists of a chain of steps based on the concept of the Stokes Structure an algebraic-analytic structure, the idea of which emerges naturally from the monodromic properties of the Gauss hypergeometric function, and which can be treated independently of the differential equations, and Fourier-like transforms adjusted to this Stokes Structure. Every step of this approach, together with all its exigencies, is illustrated by means of the non-trivial treatment of Bessel's and Weber's differential equations. It will be the aim of our future series of papers to extend this approach to matrix differential equations. It is our great pleasure to publish this series of papers in our home town and to dedicate it to the memory of our dear teacher, Naum Il'ich Akhiezer, who taught us the basic knowledge of the theory of transcendental functions and inculcated in us the taste and the love for this theory

The Stokes structure in asymptotic analysis 2: from differential equation to Stokes structure

This is the second in a series of four papers entitled 'The Stokes structure in asymptotic analysis'. We present a method of direct derivation of the Stokes structure s from a differential equation. We introduce and revise the related important definitions and statements using the Weber`s differential equation as an example. Our technique presented in this paper will be extended later to matrix differential equations

A new insight into Bessel's equation

The principal idea of this paper is to apply Fourier transforms to the Stokes structure rather than to the original differential equation. It will be shown below how to derive the Stokes structure directly from differential equation without any previous knowledge of Bessel or hypergeometric functions. It will be shown further how to adjust a Fourier transform for the Stokes structure and how to answer questions on the interrelation between formal and regular solutions of Bessel's equation using a respective Fourier analysis. It will be shown finally how to evaluate coefficients of the Stokes structure which yields a new insight into the Stokes phenomenon. Although the main object here is Bessel's equation our approach can be extended to more general matrix equations with many applications in areas such as spectral and scattering theory and hydrodynamics