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Full Description
This user-friendly textbook introduces complex analysis at the beginning graduate or advanced undergraduate level. Unlike other textbooks, it follows Weierstrass' approach, stressing the importance of power series expansions instead of starting with the Cauchy integral formula, an approach that illuminates many important concepts. This view allows readers to quickly obtain and understand many fundamental results of complex analysis, such as the maximum principle, Liouville's theorem, and Schwarz's lemma. The book covers all the essential material on complex analysis, and includes several elegant proofs that were recently discovered. It includes the zipper algorithm for computing conformal maps, as well as a constructive proof of the Riemann mapping theorem, and culminates in a complete proof of the uniformization theorem. Aimed at students with some undergraduate background in real analysis, though not Lebesgue integration, this classroom-tested textbook will teach the skills and intuition necessary to understand this important area of mathematics.
Contents
Preface; Prerequisites; Part I: 1. Preliminaries; 2. Analytic functions; 3. The maximum principle; 4. Integration and approximation; 5. Cauchy's theorem; 6. Elementary maps; Part II: 7. Harmonic functions; 8. Conformal maps and harmonic functions; 9. Calculus of residues; 10. Normal families; 11. Series and products; Part III: 12. Conformal maps to Jordan regions; 13. The Dirichlet problem; 14. Riemann surfaces; 15. The uniformization theorem; 16. Meromorphic functions on a Riemann surface; Appendix; Bibliography; Index.
