The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor's fancy at the time as well as extra reading material for students on their own. A large number of routine exercises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students.
此书为英文版。
Foreword
Prerequisites
PART ONE
Basle Theory
CHAPTER I
Complex Numbers and Functions
1. Definition
2. Polar Form
3. Complex Valued Functions
4. Limits and Compact Sets Compact Sets
5. Complex Differentiability
6. The Cauchy-Riemann Equations
7. Angles Under Holomorphic Maps
CHAPTER II
Power Series
1. Formal Power Series
2. Convergent Power Series
3. Relations Between Formal and Convergent Series
Sums and Products
Quotients
Composition of Series
4. Analytic Functions
5. Differentiation of Power Series
6. The Invelse and Open Mapping Theorems
7. The Local Maximum Modulus Principle
CHAPTER III
Cauchy's Theorem, First Part
1. Holomorphic Functions on Connected Sets Appendix: Connectedness
2. Integrals Oer Paths
3. Local Primitive for a Holomorphic Function
4. Ancther Description of 1he Integral Along a Path
5. The Homotopy Form of Cauchy's Theorem
6. Existence of Global Primitives. Definition of the Logarithm
7. The Local Cauchy Formula
CHAPTER IV
Winding Numbers and Cauchy's Theorem
1. The Winding Number
2. The Global Catchy Theorem Dixon's PIocf of Theorem 2.5 (Cauchy's Formula)
3. Artin's Proof
CHAPTER V
Applications 1 Cauchy's Integral Formula
1. Uniform Limits of Analytic Functions
2. Lament Series
3. Isolated Singularities
Removable Singularities
Poles
E sential Singularities
CHAPTER Vl
Calculus ot Residues
1. The Residue Formula
Residues of Differentials
2. Evaluation of Definite Integrals
Fourier Transforms
Trigonometric Integrals
Mellin Transforms
CHAPTER VII
Conlormsl Mappings
1. Schwarz Lemma
2. Analytic Automorphisms of the Dic
3. The Upper Half Plane
4. Olher Examples
5. Fractional Linear Transformations
CHAPTER VIII
Harmonic Functions
I. Definition
Application: Perpendicularity
Application: Flow Lines
2. Examples
3. Basic Prol;erties of Harmonic Functions
4. The Poisson Formula
The Poisson Integral as a Convolution
5. Construction of Harmonic Furctions
6. Appendix. Differentiating Under the Int(gral Sign
PART TWO
Geometric Function Theory
CHAPTER IX
Schwarz Reflection
t. Schwarz Reflection (by Complex Conjugation)
,2. Reflection Across Analytic Arcs
3. Application cf Schwatz Reflection
CHAPTER X
The Riemann Mapping Theorem
1. Statement of the Theorem
2. Compact Sets in Function Spces
3. Proof cf the Riemann Mapping Theorem
4. Behavior at the Boundary
CHAPTEA Rnalytic ContinuatiXl on Along Curves
1. Continuation Along a Curve
2. The Dilogarithm
3. Application lo Picard's Theorem
PART THREE
Various Analytic Topics
CHAPTER XII
Applications of the Maximum Modulus Principle and Jensen's Formula
1. Jensen's Formula
2. The Picard-Borel Theorem
3. Bounds by the Real Part, Borel-Carathrodory Theorem
4. The Use cf Three Circles and the Effect of Small Derivatives Hermite Interpolation Formula
5. Entire Functions with Rational Valves
6. The Phragmen-Lindelrf and Hadamard Theorems
CHAPTER XIII
Entire and Meromorphic Functions
1. Infinite Products
2. Weierstrass Products
3. Functions of Finite Order
4. Meromorphic Functions, Mittag-Leffler Theorem
CHAPTER XIV
Elliptic Functions
1. The Liouville Theorems
2. The Weierstrass Function
3. The Addition Theorem
4. The Sigma and Zeta Functions
CHAPTER XV
The Gamma and Zeta Functions
1. The Differentiation Lemma
2. The Gamma Function
Weierstrass Product
The Gauss Multiplication Formula (Distribution Relation)
The (Other) Gauss Formula
The Mellin Transform
The Starling Formula
Proof of Starling's Formula
3. The Lerch Formula
4. Zeta Functions
CHAPTER XVl
The Prime Number Theorem
1. Basic Analytic Properties of the Zeta Function
2. The Main Lemma and its Application
3. Proof of the Main Lemma
Appenflix
1. Summation by Parts and Non-Absolute Convergence
2. Difference Equations
3. Analytic Differential Equations
4. Fixed Points of a Fractional Linear Transformation
5. Cauchy's Formula for C Functions
6. Cauchy's Theorem for Locally Integrable Vector Fields
Bibliography
Index