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**Asymptotic methods for linear ordinary differential equations. (Asimptoticheskie metody dlya linejnykh obyknovennykh differential’nykh uravnenij).**
*(Russian)*
Zbl 0538.34001

Spravochnaya Matematicheskaya Biblioteka. Moskva: “Nauka” Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury. 352 p. R. 1.50 (1983).

The aim of this book is to provide fundamental results from the asymptotic theory of ordinary differential equations (ODE) and systems in the linear case. The author considers both expansions of solutions of equations containing a small parameter connected with higher derivatives and for large values of argument. It is worth while to mention that the asymptotic methods for ODE play a significant role in many applications in physics (especially in quantum mechanics) and in engineering. The author pays much attention to such applications and illustrates the presented theory by several appropriate examples.

The book consists of five chapters. Chapter 1 has an introductory character and contains mainly the classical material: Cauchy’s theorem on analytic solutions of ODE and classification (and properties) of regular and singular points. Some aspects of nonclassical material are also presented, for example transformation of boundary conditions from singular points into regular ones and so on. Chapter 2 is devoted to the linear ODE of second order on real line. First the author considers the asymptotic theory for the equation \(y''-\lambda^ 2q(x)=0\) (where \(\lambda>0\) is a large parameter) and next continues his study for more complicated forms of linear ODE of the second order involving parameters and systems of such equations. The main point investigated here is the so-called WKB method.

In Chapter 3 the author investigates the linear ODE of the second order on the complex plane. He discusses similar facts as in Chapter 2 but inasmuch as the theory ODE on complex plane has a different character then additionally the following facts are considered: asymptotic properties of eigenvalues of the operator \(-y''+\lambda^ 2q(x)\), selfadjoint and nonselfadjoint problems, the Sturm-Liouville equation with a periodic potential and others. Chapter 4 contains the discussion of the concept of a turning point for linear ODE of the second order both in a real and in a complex case. In the last Chapter, 5, equations and systems of an arbitrary order are investigated. The bibliography contains 116 items and covers the efforts of mathematicians working in the asymptotic theory for ODE. The book contains the current results from the asymptotic theory presented in a rather concise form. In this regard the book may be mainly recommended for specialists working in the asymptotic theory of ODE and using in their research the methods of this theory. But with suitable supplementation, it could provide the foundation for a good modern introduction to the study of asymptotic methods in the theory of ODE and their various applications.

The book consists of five chapters. Chapter 1 has an introductory character and contains mainly the classical material: Cauchy’s theorem on analytic solutions of ODE and classification (and properties) of regular and singular points. Some aspects of nonclassical material are also presented, for example transformation of boundary conditions from singular points into regular ones and so on. Chapter 2 is devoted to the linear ODE of second order on real line. First the author considers the asymptotic theory for the equation \(y''-\lambda^ 2q(x)=0\) (where \(\lambda>0\) is a large parameter) and next continues his study for more complicated forms of linear ODE of the second order involving parameters and systems of such equations. The main point investigated here is the so-called WKB method.

In Chapter 3 the author investigates the linear ODE of the second order on the complex plane. He discusses similar facts as in Chapter 2 but inasmuch as the theory ODE on complex plane has a different character then additionally the following facts are considered: asymptotic properties of eigenvalues of the operator \(-y''+\lambda^ 2q(x)\), selfadjoint and nonselfadjoint problems, the Sturm-Liouville equation with a periodic potential and others. Chapter 4 contains the discussion of the concept of a turning point for linear ODE of the second order both in a real and in a complex case. In the last Chapter, 5, equations and systems of an arbitrary order are investigated. The bibliography contains 116 items and covers the efforts of mathematicians working in the asymptotic theory for ODE. The book contains the current results from the asymptotic theory presented in a rather concise form. In this regard the book may be mainly recommended for specialists working in the asymptotic theory of ODE and using in their research the methods of this theory. But with suitable supplementation, it could provide the foundation for a good modern introduction to the study of asymptotic methods in the theory of ODE and their various applications.

Reviewer: J.Banas

### MSC:

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

34E05 | Asymptotic expansions of solutions to ordinary differential equations |

34A30 | Linear ordinary differential equations and systems |

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

34M99 | Ordinary differential equations in the complex domain |

34E20 | Singular perturbations, turning point theory, WKB methods for ordinary differential equations |

34E99 | Asymptotic theory for ordinary differential equations |