Preface

1. The Complex Numbers

Introduction

1.1. The Field of Complex Numbers

1.2. The Complex Plane

1.3. Topological Aspects of the Complex Plane

1.4. Stereographic Projection; The Point at Infinity

Exercises

2. Functions of the Complex Variable z

Introduction

2.1. Analytic Polynomials

2.2. Power Series

2.3. Differentiability and Uniqueness of Power Series

Exercises

3. Analytic Functions

3.1. Analyticity and the Cauchy-Riemann Equations

3.2. The Functions e^{z}, sin z, cos z

Exercises

4. Line Integrals and Entire Functions

Introduction

4.1. Properties of the Line Integral

4.2. The Closed Curve Theorem for Entire Functions

Exercises

5. Properties of Entire Functions

5.1. The Cauchy Integral Formula and Taylor Expansion for Entire Functions

5.2. Liouville Theorems and the Fundamental Theorem of Algebra

Exercises

6. Properties of Analytic Functions

Introduction

6.1. The Power Series Representation for Functions Analytic in a Disc

6.2. Analyticity in an Arbitrary Open Set

6.3. The Uniqueness, Mean-Value, and Maximum-Modulus Theorems

Exercises

7. Further Properties of Analytic Functions

7.1. The Open Mapping Theorem; Schwarz’ Lemma

7.2. The Converse of Cauchy’s Theorem: Morera’s Theorem; The Schwarz Reflection Principle

Exercises

8. Simply Connected Domains

8.1. The General Cauchy Closed Curve Theorem

8.2. The Analytic Function Log z

Exercises

9. Isolated Singularities of an Analytic Function

9.1. Classification of Isolated Singularities; Riemann’s Principle and the Casorati-Weierstrass Theorem

9.2. Laurent Expansions

Exercises

10. The Residue Theorem

lO.1. Winding Numbers and the Cauchy Residue Theorem

lO.2. Applications of the Residue Theorem

Exercises

11. Applications of The Residue Theorem to the Evaluation of Integrals and Sums

Introduction

11.1. Evaluation of Definite Integrals by Contour Integral Techniques

11.2. Application of Contour Integral Methods to Evaluation and Estimation of Sums

Exercises

12. Further Contour Integral Techniques

12.1. Shifting the Contour of Integration

12.2. An Entire Function Bounded in Every Direction

Exercises

13. Introduction to Conformal Mapping

13.1. Conformal Equivalence

13.2. Special Mappings

Exercises

14. The Riemann Mapping Theorem

14.1. Conformal Mapping and Hydrodynamics

14.2. The Riemann Mapping Theorem

Exercises

15. Maximum-Modulus Theorems for Unbounded Domains

15.1. A General Maximum-Modulus Theorem

15.2. The Phragmén-Lindelöf Theorem

Exercises

16. Harmonic Functions

16.1. Poisson Formulae and the Dirichlet Problem

16.2. Liouville Theorems for **Re** f; Zeroes of Entire Functions of Finite Order

Exercises

17. Different Forms of Analytic Functions

Introduction

17.1. Infinite Products

17.2. Analytic Functions Defined by Definite Integrals

Exercises

18. Analytic Continuation; The Gamma and Zeta Functions

Introduction

18.1. Power Series

18.2. The Gamma and Zeta Functions

Exercises

19. Applications to Other Areas of Mathematics

Introduction

19.1. A Partition Problem

19.2. An Infinite System of Equations

19.3. A Variation Problem

19.4. The Fourier Uniqueness Theorem

19.5. The Prime-Number Theorem

Exercises

Appendices

1. A Note on Simply Connected Regions

II. Circulation and Flux as Contour Integrals

III. Steady-State Temperatures; The Heat Equation

IV. Tchebychev Estimates

Answers

Bibliography

Index