Author	Andersson, Mats. author
Title	Topics in Complex Analysis [electronic resource] / by Mats Andersson
Imprint	New York, NY : Springer New York, 1997
Connect to	http://dx.doi.org/10.1007/978-1-4612-4042-6
Descript	VIII, 157 p. online resource

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 asยญ sumes that the reader has previous knowledge of basic real analysis. This makes it possible to follow a rather quick route through the most fundamenยญ 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 analยญ ysis, while at the same time treating classical complex function theory

Preliminaries -- ยง1. Notation -- ยง2. Some Facts -- 1. Some Basic Properties of Analytic Functions -- ยง1. Definition and Integral Representation -- ยง2. Power Series Expansions and Residues -- ยง3. Global Cauchy Theorems -- 2. Properties of Analytic Mappings -- ยง1. Conformal Mappings -- ยง2. The Riemann Sphere and Projective Space -- ยง3. Univalent Functions -- ยง4. Picardโ{128}{153}s Theorems -- 3. Analytic Approximation and Continuation -- ยง1. Approximation with Rationals -- ยง2. Mittag-Lefflerโ{128}{153}s Theorem and the Inhomogeneous Cauchy-Riemann Equation -- ยง3. Analytic Continuation -- ยง4. Simply Connected Domains -- ยง5. Analytic Functionals and the Fourier-Laplace Transform -- ยง6. Mergelyanโ{128}{153}s Theorem -- 4. Harmonic and Subharmonic Functions -- ยง1. Harmonic Functions -- ยง2. Subharmonic Functions -- 5. Zeros, Growth, and Value Distribution -- ยง1. Weierstrassโ{128}{153} Theorem -- ยง2. Zeros and Growth -- ยง3. Value Distribution of Entire Functions -- 6. Harmonic Functions and Fourier Series -- ยง1. Boundary Values of Harmonic Functions -- ยง2. Fourier Series -- 7. IF Spaces -- ยง1. Factorization in Hp Spaces -- ยง2. Invariant Subspaces of H2 -- ยง3. Interpolation of H? -- ยง4. Carleson Measures -- 8. Ideals and the Corona Theorem -- ยง1. Ideals in A(?) -- ยง2. The Corona Theorem -- 9. H1 and BMO -- ยง1. Bounded Mean Oscillation -- ยง2. The Duality of H1 and BMO -- List of Symbols