Synopses & Reviews
This book treats several topics on the global analysis of linear differential equations in the complex domain, using original methods of analysis developed by K. Okubo and the present author. For example, global analysis of linear differential equations is closely related to that of linear difference equations. Through the asymptotic behaviour of solutions of linear difference equations, connection problems for hypergeometric systems can be solved, i.e. Fuchsian differential equations and Stokes phenomenon for Birkhoff systems. Furthermore, the calculation of monodromy groups is explained in detail. Audience: This volume will be of interest to researchers and graduate students whose work involves ordinary differential equations, special functions, finite difference, functional equations, approximations and expansions, and symbolic/algebraic manipulation.
Synopsis
Since the initiative works for global analysis of linear differential equations by G.G. Stokes and B. Riemann in 1857, the Airy function and the Gauss hypergeometric function became the most important and the greatest practical special functions, which have a variety of applications to mathematical science, physics and engineering. The cffcctivity of these functions is essentially due to their "behavior in the large" . For instance, the Airy function plays a basic role in the asymptotic analysis of many functions arising as solutions of differential equations in several problems of applied math ematics. In case of the employment of its behavior, one should always pay attention to the Stokes phenomenon. On the other hand, as is well-known, the Gauss hypergeometric function arises in all fields of mathematics, e.g., in number theory, in the theory of groups and in analysis itself. It is not too much to say that all power series are special or extended cases of the hypergeometric series. For the full use of its properties, one needs connection formulas or contiguous relations."
Synopsis
This book treats several topics on the global analysis oflinear differential equations in the complex domain, using originalmethods of analysis developed by K. Okubo and the present author. Forexample, global analysis of linear differential equations is closelyrelated to that of linear difference equations. Through the asymptoticbehaviour of solutions of linear difference equations, connectionproblems for hypergeometric systems can be solved, i.e. Fuchsiandifferential equations and Stokes phenomenon for Birkhoff systems.Furthermore, the calculation of monodromy groups is explained indetail."Audience: " This volume will be of interest to researchers andgraduate students whose work involves ordinary differential equations, special functions, finite difference, functional equations, approximations and expansions, and symbolic/algebraic manipulation.
Description
Includes bibliographical references (p. 515-522) and index.
Table of Contents
Preface. 1. Preparations. 2. Gauss and Airy Equations. 3. Linear Differential Equations. 4. Reduction Problems. 5. Monodromy Groups for Hypergeometric Systems. 6. Connection Problem for Hypergeometric Systems. 7. Stokes Phenomenon. Bibliography. Index.