Generalized analytic functions on Riemann surfaces / Yuri L. Rodin

The theory of generalized analytic functions (pseudo-analytic functions) as introduced by L. Bers and I. N. Vekua originated in the interest in generalizations of the idea of analyticity and in problems in partial differential equations. Only a few years later the Riemann boundary problem was studied on Riemann surfaces by A. Grothendieck, W. Koppelman, and others . When relations of algebraic function theory to the theory of singular integral operators and the classification problem of vector bundles over Riemann surfaces found physical applications, the study of generalized analytic functions on Riemann surfaces was stimulated. One of the active contributors in this field since about 1960 is the author of this book, and for the first time here he gives a concise and systematic survey of results in this theory.
In order to give a rough idea of the contents, we note the following chapter headings: the Dolbeault and Riemann-Roch theorems, linear integral equations connected with generalized analytic functions, the Riemann boundary problem, nonlinear aspects of generalized analytic function theory, some generalizations and applications. This is followed by an appendix on cohomologies with coefficients in sheaves and about 100 references.
The book will serve the growing interest in the theory of generalized analytic functions and is a helpful guide to the literature, for mathematicians as well as for physicists (MathSciNet)