This book is intended as a textbook for a first course in the theory offunctions of one complex variable for students who are mathematicallymature enough to understand and execute arguments. The actual pre-requisites for reading this book are quite minimal；not much more than astiff course in basic calculus and a few facts about partial derivatives. Thetopics from advanced calculus that are used are proved in detail.

Preface

Ⅰ. The Complex Number System

 1. The real numbers

 2. The field of complex numbers

 3. The complex plane

 4. Polar representation and roots of complex numbers

 5. Lines and half planes in the complex plane

 6. The extended plane and its spherical representation

Ⅱ. Metric Spaces and the Topology of C

 1. Definition and examples of metric spaces

 2. Connectedness

 3. Sequences and completeness

 4. Compactness

 5. Continuity

 6. Uniform convergence

Ⅲ. Elementary Properties and Examples of Analytic Functions

 1. Power series

 2. Analytic functions

 3. Analytic functions as mappings， M6bius transformations

Ⅳ. Complex Integration

 1.Riemann-Stieltjes integrals

 2.Power series representation of analytic functions

 3.Zeros of an analytic function

 4.The index of a closed curve

 5.Cauchy's Theorem and Integral Formula

 6.The homotopic version of Cauchy's Therorem and simple connectivity

 7.Counting zeros;the Open Mapping Theorem

 8.Goursat's Theorem

Ⅴ.Singularities

 1.Classification of singularities

 2.Residues

 3.The Argument Principle

Ⅵ.The Maximum Modulus Theorem

 §1.The Maximum Principle 

 §2.Schwarz’S Lemma   

 §3.Convex functions and Hadamard’S Three Circles Theorem

 §4.Phragm6n-Lindel6f Theorem  

Ⅶ.Compactness and Convergence in the

 Space of Analytic Functions

 §1.The space of continuous functions C(G，Q) 

 §2.Spaces of analytic functions   

 §3.Spaces of meromorphic functions  

 §4.The Riemann Mapping Theorem  

 §5.Weierstrass Factorization Theorem 

 §6.Factorization of the sine function   

 §7.The gamma function    

 §8.The Ricmann zeta function  

Ⅷ.Runge’S Theorem

 §1.Runge’S Theorem   

 §2.Simple connectedness  

 §3.Mittag·Leffer’s Theorem   

Ⅸ.Analytic Continuation and Riemann SurfaCeS

 §1.Schwarz Reflection Principle 

 §2.Analytic Continuation Along A Path  

 §3.Mondromy Theorem   

 §4.Topological Spaces and Neighborhood Systems 

 §5.The Sheaf of Germs of Analytic Functions on an Open Set

 §6.Analytic ManifoIds   

 §7.Covering spaces    

Ⅹ.Harmonic Functions

 §1.Basic Properties of harmonic functions 

 §2.Harmonic functions on a disk 

 §3.Subharmonic and SUPerharmonic functions 

 §4.The Dirichlet Problem   

 §5.Gregn’s Functions   

Ⅺ. Entire Functions

 §1.Jensen’S Formula   

 §2.The genus and order of an entire function 

 §3.Hadamard Factorization Theorem 

Ⅻ.The Range of an Analytic Function

 §1.Bloch’S Theorem  

 §2.The Little Picard Theorem  

 §3.Schottky’S Theorem   

 §4.The Great Picard Theorem  

Appendix A：Calculus for Complex Valued Functions on an Interval

Appendix B：Suggestions for Further Study and

Bibliographical Notes 

References   

Index    

List of Symbols