Linear Differential Equations in the Complex Domain

This book provides a detailed introduction to recent developments in the theory of linear differential systems and integrable total differential systems. Starting from the basic theory of linear ordinary differential equations and integrable systems, it proceeds to describe Katz theory and its applications, extending it to the case of several variables. In addition, connection problems, deformation theory, and the theory of integral representations are comprehensively covered. Complete proofs are given, offering the reader a precise account of the classical and modern theory of linear differential equations in the complex domain, including an exposition of Pfaffian systems and their monodromy problems. The prerequisites are a course in complex analysis and the basics of differential equations, topology and differential geometry. This book will be useful for graduate students, specialists in differential equations, and for non-specialists who want to usedifferential equations.

Presents the first systematic account of the classical and modern theory of complex linear differential equations, covering results previously scattered throughout the literature Gives a clear introduction to advanced (non-standard) topics and recent developments in the theory Offers a detailed presentation with many examples and full proofs, making it accessible to graduate students and non-specialists Provides motivation and historical background for each topic covered

Zusammenfassung

"The author has made many contributions to the related global theory of Fuchsian equations, including hypergeometric-type equations and Okubo systems. In this monograph, the author leads the reader to the forefront of this field." (Shun Shimomura, Mathematical Reviews, March, 2022)

"The main purpose of this interesting book is an introduction to recent developments in the theory of linear ordinary differential systems and linear completely integrable total differential systems. ... The book is designed to be suitable for use as a primary textbook in an advanced graduate course or as a supplementary source for beginning graduate courses. ... The book contains numerous examples, which can be useful as a reference for researchers." (Valentine Tyshchenko, zbMATH 1460.34002, 2021)

Inhalt

• Introduction. - Part I Ordinary Differential Equations. - Scalar Differential Equations and Systems of Differential Equations. - Analysis at a Regular Point. - Regular Singular Points. - Monodromy. - Connection Problem. - Fuchsian Differential Equations. - Deformation Theory. - Integral Representations of Solutions of Euler Type. - Irregular Singular Points. - Part II Completely Integrable Systems. - Linear Pfaffian Systems and Integrability Condition. - Regular Singularity. - Monodromy Representations. - Middle Convolution.