Complex Analysis

This unusual and lively textbook offers a clear and intuitive approach to the classical and beautiful theory of complex variables. With very little dependence on advanced concepts from several-variable calculus and topology, the text focuses on the authentic complex-variable ideas and techniques. Accessible to students at their early stages of mathematical study, this full first year course in complex analysis offers new and interesting motivations for classical results and introduces related topics stressing motivation and technique. Numerous illustrations, examples, and now 300 exercises, enrich the text. Students who master this textbook will emerge with an excellent grounding in complex analysis, and a solid understanding of its wide applicability.

Table of Contents

1. The Complex Numbers

2. Functions of the Complex Variable

3. Analytic Functions

4. Line Integrals and Entire Functions

5. Properties of Entire Functions

6. Properties of Analytic Functions

7. Further Properties of Analytic Functions

8. Simply Connected Domains

9. Isolated Singularities of an Analytic Function

10. The Residue Theorem

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

12. Further Contour Integral Techniques

13. Introduction to Conformal Mapping

14. The Riemann Mapping Theorem

15. Maximum-Modulus Theorems for Unbounded Domains

16. Harmonic Functions

17. Different Forms of Analytic Functions

18. Analytic Continuation; The Gamma and Zeta Functions

19. Applications to Other Areas of Mathematics

About the Authors

Joseph Bak is the Assistant Chair of the Mathematics department at The City College of New York. Joseph Bak’s primary area of research is approximation theory.

Donald J. Newman (July 27, 1930 – March 28, 2007) was a champion problem solver. His mathematical specialties included complex analysis, approximation theory and number theory. His career included posts as a Professor of Mathematics at MIT, Brown University, Yeshiva University, Temple University and a distinguished chair at Bar Ilan University in Israel. His publications include 150 papers and five books.