Topics in Complex Analysis

This book is an outgrowth of lectures given on several occasions at Chalmers University of Technology and Goteborg University during the last ten years. As opposed to most introductory books on complex analysis, this one as­ sumes that the reader has previous knowledge of basic real analysis. This makes it possible to follow a rather quick route through the most fundamen­ tal material on the subject in order to move ahead to reach some classical highlights (such as Fatou theorems and some Nevanlinna theory), as well as some more recent topics (for example, the corona theorem and the HI_ BMO duality) within the time frame of a one-semester course. Sections 3 and 4 in Chapter 2, Sections 5 and 6 in Chapter 3, Section 3 in Chapter 5, and Section 4 in Chapter 7 were not contained in my original lecture notes and therefore might be considered special topics. In addition, they are completely independent and can be omitted with no loss of continuity. The order of the topics in the exposition coincides to a large degree with historical developments. The first five chapters essentially deal with theory developed in the nineteenth century, whereas the remaining chapters contain material from the early twentieth century up to the 1980s. Choosing methods of presentation and proofs is a delicate task. My aim has been to point out connections with real analysis and harmonic anal­ ysis, while at the same time treating classical complex function theory.

This book provides a concise treatment of topics in complex analysis, suitable for a one-semester course. It is an outgrowth of lectures given by the author over the last ten years at the University of Goteborg and Chalmers University of Technology. While treating classical complex function theory, the author emphasizes connections to real and harmonic analysis, and presents general tools that might be useful in other areas of analysis. The book introduces all of the basic ideas in beginning complex analysis and still has time to reach many topics near the frontier of the subject. The reader is expected to have an understanding of basic integration theory and functional analysis. Many exercises illustrate and sharpen the theory, and extended exercises give the reader an active part in complementing the material presented in the text.