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.
Author(s): Donald E. Marshall
Series: Cambridge Mathematical Textbooks
Publisher: Cambridge University Press
Year: 2019
Language: English
Commentary: True PDF
Pages: 286
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
