Donald E. MarshallCambridge University Press, 4/18/2019EAN 9781107134829, ISBN10: 110713482XHardcover, 308 pages, 26.1 x 18.2 x 1.8 cmLanguage: EnglishThis 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.PrefacePrerequisitesPart I1. Preliminaries2. Analytic functions3. The maximum principle4. Integration and approximation5. Cauchy's theorem6. Elementary mapsPart II7. Harmonic functions8. Conformal maps and harmonic functions9. Calculus of residues10. Normal families11. Series and productsPart III12. Conformal maps to Jordan regions13. The Dirichlet problem14. Riemann surfaces15. The uniformization theorem16. Meromorphic functions on a Riemann surfaceAppendixBibliographyIndex.
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