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.

Inhalt
Preliminaries.- §1. Notation.- §2. Some Facts.- 1. Some Basic Properties of Analytic Functions.- §1. Definition and Integral Representation.- §2. Power Series Expansions and Residues.- §3. Global Cauchy Theorems.- 2. Properties of Analytic Mappings.- §1. Conformal Mappings.- §2. The Riemann Sphere and Projective Space.- §3. Univalent Functions.- §4. Picard's Theorems.- 3. Analytic Approximation and Continuation.- §1. Approximation with Rationals.- §2. Mittag-Leffler's Theorem and the Inhomogeneous Cauchy-Riemann Equation.- §3. Analytic Continuation.- §4. Simply Connected Domains.- §5. Analytic Functionals and the Fourier-Laplace Transform.- §6. Mergelyan's Theorem.- 4. Harmonic and Subharmonic Functions.- §1. Harmonic Functions.- §2. Subharmonic Functions.- 5. Zeros, Growth, and Value Distribution.- §1. Weierstrass' Theorem.- §2. Zeros and Growth.- §3. Value Distribution of Entire Functions.- 6. Harmonic Functions and Fourier Series.- §1. Boundary Values of Harmonic Functions.- §2. Fourier Series.- 7. IF Spaces.- §1. Factorization in Hp Spaces.- §2. Invariant Subspaces of H2.- §3. Interpolation of H?.- §4. Carleson Measures.- 8. Ideals and the Corona Theorem.- §1. Ideals in A(?).- §2. The Corona Theorem.- 9. H1 and BMO.- §1. Bounded Mean Oscillation.- §2. The Duality of H1 and BMO.- List of Symbols.