This book is an outgrowth of lectures given at several occasions at the University of Göteborg and Chalmers University of Technology during the last ten years. Contrary to most introductory texts on complex analysis, it preassumes knowledge of basic analysis. This makes it possible to move rather quickly through the most fundamental material and to reach within a one-semester course some classical highlights such as Fatou theorems and some Nevanlinna theory, as well as more recent topics, for example the corona theorem and the H1-BMO duality.
Author(s): Mats Andersson
Series: Universitext / Universitext: Tracts in Mathematics
Publisher: Springer
Year: 1997
Language: English
Pages: 167
Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 6
Contents......Page 8
1. Notation......Page 10
2. Some Facts......Page 11
1. Definition and Integral Representation......Page 14
2. Power Series Expansions and Residues......Page 21
3. Global Cauchy Theorems......Page 27
1. Conformal Mappings......Page 37
2. The Riemann Sphere and Projective Space......Page 42
3. Univalent Functions......Page 44
4. Picard's Theorems......Page 47
1. Approximation with Rationals......Page 55
2. Mittag-Leffler's Theorem and the Inhomogeneous Cauchy-Riemann Equation......Page 57
3. Analytic Continuation......Page 60
4. Simply Connected Domains......Page 62
5. Analytic Functionals and the Fourier-Laplace Transform......Page 64
6. Mergelyan's Theorem......Page 67
1. Harmonic Functions......Page 76
2. Subharmonic Functions......Page 80
1. Weierstrass' Theorem......Page 91
2. Zeros and Growth......Page 94
3. Value Distribution of Entire Functions......Page 97
1. Boundary Values of Harmonic Functions......Page 106
2. Fourier Series......Page 113
1. Factorization in H^p Spaces......Page 121
2. Invariant Subspaces of H^2......Page 125
3. Interpolation of H?......Page 127
4. Carleson Measures......Page 130
1. Ideals in A(12)......Page 139
2. The Corona Theorem......Page 140
1. Bounded Mean Oscillation......Page 150
2. The Duality of H' and BMO......Page 155
Bibliography......Page 160
List of Symbols......Page 162
Index......Page 164
