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 (e.g., Leibniz's rule for differ-entiating under the integral sign) 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.Compactness and Convergence in the
Ⅶ.Runge's Theorem
Ⅷ.Analytic Continuation and Riemann Surfaces
Ⅸ.Harmonic Functions
Ⅹ.Entire Functions
Ⅺ.The Range of an Analytic Function
Appendix A:Calculus for Complex Complex Valued Functions of and Interval
Appendix B:Suggestions for Further Study and Bibliographical Notes
References
Index
List of Symbols