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AMS Chelsea Publishing
1981; 309 pp; hardcover
first AMS printing 2002
List Price: US$44
Member Price: US$39.60
Order Code: CHEL/313.H
This well-known book is a self-contained treatment of the classical theory of abstract Riemann surfaces. The first five chapters cover the requisite function theory and topology for Riemann surfaces. The second five chapters cover differentials and uniformization. For compact Riemann surfaces, there are clear treatments of divisors, Weierstrass points, the Riemann-Roch theorem and other important topics.
Springer's book is an excellent text for an introductory course on Riemann surfaces. It includes exercises after each chapter and is illustrated with a beautiful set of figures.
"Written with unusual clearness. As in the Introduction, which outlines the whole book, similar [outlines] appear in each chapter ... a modern treatment in a self-contained manner with a minimum assumption of knowledge. He is most successful in this magnificent project ... It is highly recommended."
-- American Mathematical Monthly
"The book is written specifically with graduate (and advanced undergraduate) students in mind. There are no prerequisites beyond standard first courses in complex variables, real variables, and algebra. What is needed of topology and Hilbert space theory is derived from the beginning. Concepts and theorems are illuminated by examples and excellent figures, proofs are clarified by heuristic remarks, and the inventiveness of even the good student is challenged by a well chosen problem collection. The style, while very readable, never becomes "insultingly simple" and even the specialist can derive pleasure from reviewing basic material in a well-organized form."
-- Mathematical Reviews
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