The subject matter loosely called “Riemann surface theory” has been the starting point for the development of topology, functional analysis, modern algebra, and any one of a dozen recent branches of mathematics; it is one of the most valuable bodies of knowledge within mathematics for a student to learn.Professor Cohn's lucid and insightful book presents an ideal coverage of the subject in five pans. Part I is a review of complex analysis analytic behavior, the Riemann sphere, geometric constructions, and presents (as a review) a microcosm of the course. The Riemann manifold is introduced in Part II and is examined in terms of intuitive physical and topological technique in Part III. In Part IV the author shows how to define real functions on manifolds analogously with the algebraic and analytic points of view outlined here. The exposition returns in Part V to the use of a single complex variable z. As the text is richly endowed with problem material — 344 exercises — the book is perfect for self-study as well as classroom use.Harvey Cohn is well-known in the mathematics profession for his pedagogically superior texts, and the present book will be of great interest not only to pure and applied mathematicians, but also engineers and physicists. Dr. Cohn is currently Distinguished Professor of Mathematics at the City University of New York Graduate Center.