Synopses & Reviews
This book is devoted to the asymptotic theory of differential equations. Asymptotic theory is an independent and important branch of mathematical analysis that began to develop at the end of the 19th century. Asymptotic methods' use of several important phenomena of nature can be explained. The main problems considered in the text are based on the notion of an asymptotic space, which was introduced by the author in his works. Asymptotic spaces for asymptotic theory play analogous roles as metric spaces for functional analysis. It allows one to consider many (seemingly) miscellaneous asymptotic problems by means of the same methods and in a compact general form. The book contains the theoretical material and general methods of its application to many partial problems, as well as several new results of asymptotic behavior of functions, integrals, and solutions of differential and difference equations. Audience: The material will be of interest to mathematicians, researchers, and graduate students in the fields of ordinary differential equations, finite differences and functional equations, operator theory, and functional analysis.
Synopsis
The asymptotic theory deals with the problern of determining the behaviour of a function in a neighborhood of its singular point. The function is replaced by another known function ( named the asymptotic function) close (in a sense) to the function under consideration. Many problems of mathematics, physics, and other divisions of natural sci- ence bring out the necessity of solving such problems. At the present time asymptotic theory has become an important and independent branch of mathematical analysis. The present consideration is mainly based on the theory of asymp- totic spaces. Each asymptotic space is a collection of asymptotics united by an associated real function which determines their growth near the given point and (perhaps) some other analytic properties. The main contents of this book is the asymptotic theory of ordinary linear differential equations with variable coefficients. The equations with power order growth coefficients are considered in detail. As the application of the theory of differential asymptotic fields, we also consider the following asymptotic problems: the behaviour of explicit and implicit functions, improper integrals, integrals dependent on a large parameter, linear differential and difference equations, etc .. The obtained results have an independent meaning. The reader is assumed to be familiar with a comprehensive course of the mathematical analysis studied, for instance at mathematical departments of universities. Further necessary information is given in this book in summarized form with proofs of the main aspects.
Table of Contents
1. Introduction.
2. Metric spaces.
3. Asymptotic spaces.
4. Asymptotic behaviour of functions.
5. Power order growth functions on the positive semi-axis.
6. Power order growth functions of the complex argument.
7. Integrals.
8. Linear differential equations.
9. General asymptotic properties of linear differential equations.
10. Linear differential equations with power order growth coefficients on the positive semi-axis.
11. Linear differential equations in singular cases on the positive semi-axis.
12. Linear differential equations in a sector of the complex plane.
13. Linear differential equations with power-logarithmic coefficients.
14. Linear difference equations. General theory.
15. Asymptotic behaviour of solutions of linear difference equations.
16. Supplement. Bibliography. List of symbols. Index.