Complex analysis is one of the most central subjects in mathematics. It is compelling and rich in its own right, but it is also remarkably useful in a wide variety of other mathematical subjects, both pure and applied. This book is different from others in that it treats complex variables as a direct development from multivariable real calculus. As each new idea is introduced, it is related to the corresponding idea from real analysis and calculus. The text is rich with examples and exercises that illustrate this point.

The authors have systematically separated the analysis from the topology, as can be seen in their proof of the Cauchy theorem. The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem, and the Bergman kernel. The authors also treat $H^p$ spaces and Painlevé''''s theorem on smoothness to the boundary for conformal maps.

This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors'''' expertise both as mathematicians and as expositors

Robert E. Greene, University of California, Los Angeles, CA, and Steven G. Krantz, Washington University, St. Louis, MO.

1. Fundamental Concepts
2. Complex Line Integrals
3. Applications of the Cauchy Integral
4. Meromorphic Functions and Residues
5. The Zeros of a Holomorphic Function
6. Holomorphic Functions as Geometric Mappings
7. Harmonic Functions
8. Infinite Series and Products
9. Applications of Infinite Sums and Products
10. Analytic Continuation
11. Topology
12. Rational Approximation Theory
13. Special Classes of Holomorphic Functions
14. Hilbert Spaces of Holomorphic Functions, the Bergman Kernel, and Biholomorphic Mappings
15. Special Functions
16. The Prime Number Theorem