Contents Preface xiii Chapter 1. Selected Problems in One Complex Variable 1 §1.1. Preliminaries 2 §1.2. A Simple Problem 2 §1.3. Partitions of Unity 4 §1.4. The Cauchy-Riemann Equations 7 §1.5. The Proof of Proposition 1.2.2 10 §1.6. The Mittag-Leffler and Weierstrass Theorems 12 §1.7. Conclusions and Comments 16 Exercises 18 Chapter 2. Holomorphic Functions of Several Variables 23 §2.1. Cauchy's Formula and Power Series Expansions 23 §2.2. Hartog's Theorem 26 §2.3. The Cauchy-Riemann Equations 29 §2.4. Convergence Theorems 29 §2.5. Domains of Holomorphy 31 Exercises 35 Chapter 3. Local Rings and Varieties 37 §3.1. Rings of Germs of Holomorphic Functions 38 53.2. Hilbert's Basis Theorem 39
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