Complex Analysis (Cambridge Mathematical Textbooks)

Description

This user-friendly textbook introduces complex analysis at the beginning graduate or advanced undergraduate level. Unlike other textbooks, it follows Weierstrass' approach, stressing the importance of power series expansions instead of starting with the Cauchy integral formula, an approach that illuminates many important concepts. This view allows readers to quickly obtain and understand many fundamental results of complex analysis, such as the maximum principle, Liouville's theorem, and Schwarz's lemma. The book covers all the essential material on complex analysis, and includes several elegant proofs that were recently discovered. It includes the zipper algorithm for computing conformal maps, as well as a constructive proof of the Riemann mapping theorem, and culminates in a complete proof of the uniformization theorem. Aimed at students with some undergraduate background in real analysis, though not Lebesgue integration, this classroom-tested textbook will teach the skills and intuition necessary to understand this important area of mathematics. Contents List of Figures Preface Prerequisites PART I 1 Preliminaries 1.1 Complex Numbers 1.2 Estimates 1.3 Stereographic Projection 1.4 Exercises 2 Analytic Functions 2.1 Polynomials 2.2 Fundamental Theorem of Algebra and Partial Fractions 2.3 Power Series 2.4 Analytic Functions 2.5 Elementary Operations 2.6 Exercises 3 The Maximum Principle 3.1 The Maximum Principle 3.2 Local Behavior 3.3 Growth on Cand D 3.4 Boundary Behavior 3.5 Exercises 4 Integration and Approximation 4.1 Integration on Curves 4.2 Equivalence of Analytic and Holomorphic 4.3 Approximation by Rational Functions 4.4 Exercises 5 Cauchy’s Theorem 5.1 Cauchy’s Theorem 5.2 Winding Number 5.3 Removable Singularities 5.4 Laurent Series 5.5 The Argument Principle 5.6 Exercises 6 Elementary Maps 6.1 Linear Fractional Transformations 6.2 Exp and Log 6.3 Power Maps 6.4 The Joukovski Map 6.5 Trigonometric Functions 6.6 Constructing Conformal Maps 6.7 Exercises PART II 7 Harmonic Functions 7.1 The Mean-Value Property and the Maximum Principle 7.2 Cauchy–Riemann and Laplace Equations 7.3 Hadamard, Lindelöf and Harnack 7.4 Exercises 8 Conformal Maps and Harmonic Functions 8.1 The Geodesic Zipper Algorithm 8.2 The Riemann Mapping Theorem 8.3 Symmetry and Conformal Maps 8.4 Conformal Maps to Polygonal Regions 8.5 Exercises 9 Calculus of Residues 9.1 Contour Integration and Residues 9.2 Some Examples 9.3 Fourier and Mellin Transforms 9.4 Series via Residues 9.5 Laplace and Inverse Laplace Transforms 9.6 Exercises 10 Normal Families 10.1 Normality and Equicontinuity 10.2 Riemann Mapping Theorem Revisited 10.3 Zalcman, Montel and Picard 10.4 Exercises 11 Series and Products 11.1 Mittag-Leffler’s Theorem 11.2 Weierstrass Products 11.3 Blaschke Products 11.4 The Gamma and Zeta Functions 11.5 Exercises PART III 12 Conformal Maps to Jordan Regions 12.1 Some Badly Behaved Regions 12.2 Janiszewski’s Lemma 12.3 Jordan Curve Theorem 12.4 Carathéodory’s Theorem 12.5 Exercises 13 The Dirichlet Problem 13.1 Perron Process 13.2 Local Barriers 13.3 Riemann Mapping Theorem Again 13.4 Exercises 14 Riemann Surfaces 14.1 Analytic Continuation and Monodromy 14.2 Riemann Surfaces and Universal Covers 14.3 Deck Transformations 14.4 Exercises 15 The Uniformization Theorem 15.1 The Modular Function 15.2 Green’s Function 15.3 Simply-Connected Riemann Surfaces 15.4 Classification of All Riemann Surfaces 15.5 Exercises 16 Meromorphic Functions on a Riemann Surface 16.1 Existence of Meromorphic Functions 16.2 Properly Discontinuous Groups on C∗ and C 16.3 Elliptic Functions 16.4 Fuchsian Groups 16.5 Blaschke Products and Convergence Type 16.6 Exercises Appendix A.1 Fifteen Conditions Equivalent to Analytic A.2 Program for Color Pictures Bibliography Index