Contents Preface vii Acknowledgments xv Chapter 1. From i to z: the basics of complex analysis 1 §1.1. The field of complex numbers 1 §1.2. Holomorphic, analytic, and conformal 4 §1.3. The Riemann sphere 9 §1.4. obius transformations 11 §1.5. The hyperbolic plane and the Poincar´ e disk 15 §1.6. Complex integration, Cauchy theorems 18 §1.7. Applications of Cauchy’s theorems 23 §1.8. Harmonic functions 33 §1.9. Problems 36 Chapter 2. From z to the Riemann mapping theorem: some finer points of basic complex analysis 41 §2.1. The winding number 41 §2.2. The global form of Cauchy’s theorem 45 §2.3. Isolated singularities and residues 47 §2.4. Analytic continuation 56 §2.5. Convergence and normal families 60 §2.6. The Mittag-Leffler and Weierstrass theorems 63 §2.7. The Riemann mapping theorem 69 §2.8. Runge’s theorem and simple connectivity 74 iii
Previous Page Next Page