Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation

Research in differential equations is usually oriented toward explicit results and motivated by applications. Many clever methods have been discovered in this way, but, when problems of more fundamental difficulty arise, researchers must find something intrinsic in the mathematics itself in order to make progress. As research in topology, algebraic geometry, and functions of several complex variables have advanced, many methods useful in such fields were introduced into the study of differential equations. The main part of this book is a translation of a 1976 book originally written in Japanese. The book, focusing attention on intrinsic aspects of the subject, explores some problems of linear ordinary differential equations in complex domains. Examples of the problems discussed include the Riemann problem on the Riemann sphere, a characterization of regular singularities, and a classification of meromorphic differential equations. Since the original book was published, many new ideas have developed, such as applications of D-modules, Gevrey asymptotics, cohomological methods, k-summability, and studies of differential equations containing parameters. Five appendices, added in the present edition, briefly cover these new ideas. In addition, more than 100 references have been added. This book introduces readers to the essential facts concerning the structure of solutions of linear differential equations in the complex domain, as well as illuminating the intrinsic meaning of older results by means of more modern ideas. A useful reference for research mathematicians on various fundamental results, this book would also be suitable as a textbook in a graduate course or seminar.