Complex Analysis

This user-friendly textbook follows Weierstrass' approach to offer a self-contained introduction to complex analysis.

Donald E. Marshall (Author)

9781107134829, Cambridge University Press

Hardback, published 7 March 2019

286 pages, 68 colour illus.
26.1 x 18.2 x 1.8 cm, 0.77 kg

'Written by a skillful teacher and grand master of complex analysis, this complex analysis graduate level textbook stands out from other texts through the clarity and elegance of the arguments, the efficiency of the presentation, and the selection of advanced topics. Each of the 16 chapters ends with a carefully selected set of exercises ranging from routine to challenging, making it an excellent textbook and ideal for a first-year graduate course. Marshall's choice of beginning with power series (following Weierstrass) has the advantage of a very fast and direct approach to some of the highlights of the theory. The connection to Cauchy's integral calculus, which is the starting point of most texts, is then made through partial fractions and Runge's theorem. This makes the book an invaluable addition to the complex analysis literature.' Steffen Rohde, University of Washington

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.

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.

Subject Areas: Mathematical physics [PHU], Complex analysis, complex variables [PBKD], Real analysis, real variables [PBKB]