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Publisher:

Cambridge University Press

Publication Date:

2007

Number of Pages:

406

Format:

Hardcover

Series:

Cambridge Studies in Advanced Mathematics 107

Price:

80.00

ISBN:

9780521809375

Category:

Textbook

[Reviewed by , on ]

Michael Berg

07/25/2008

The back cover of *Complex Analysis*, by the great algebraic and differential geometer Kunihiko Kodaira (1915–1997), features the phrase, “[w]ritten by a master of the subject, this textbook will be appreciated by students and experts.” To be sure, this describes this marvelous book very well. The only complex analysis books of comparable high quality that come to mind are the famous texts by Ahlfors, by Hille, and by Saks and Zygmund. The attention to detail in Kodaira’s book is (at least) comparable to that found in these classics, and the book under review possesses a number of additional features that render it unique and therefore all but indispensable.

For example, the last three chapters of *Complex Analysis* are devoted to a very thorough treatment of Riemann surfaces along the lines of Weyl’s seminal *The Concept of a Riemann Surface (Part II)*, and includes wonderful discussions of the theory of differential forms, Dirichlet’s principle, (co)homology, and analytic functions on a closed Riemann surface, featuring the Riemann-Roch Theorem and Abel’s Theorem. *Qua* cohomology, de Rham’s Theorem (with a proof) occurs on p. 342 ff., Riemann-Roch appears on p. 382 ff., and Abel’s Theorem takes up the last three pages of the book, pp. 391–393: a very fitting climax.

Prior to all this, in the book’s fifth chapter, Kodaira provides an exceptionally accessible discussion of the Riemann mapping theorem, leading to an elegant presentation of the reflection principle. The chapter ends with Picard’s Theorem and the Schwartz-Christoffel formula.

These features certainly conspire to render Kodaira’s book not only exceptional but also a fine source for studies in complex analysis at the graduate school level. One warning, however: the exercises in the book occur (grouped by chapter) at the very end, in the space of only seven pages, so a graduate student had better look elsewhere for more extensive problem sets. But as these sets are ubiquitous this is obviously not a real detraction from the book’s quality.

Finally a few words about the more elementary material covered in *Complex Analysis*, i.e., the contents of the first four chapters. Most of this material should be within reach of a strong undergraduate, at least as far as the level of difficulty goes; but there is so much offered there that a semester isn’t enough: a year is needed to do justice to it. To wit, Kodaira begins with holomorphic functions (in the old style; f´ is supposed to be continuous), hits Cauchy theory very, very hard (including a discussion of homology), presents conformal mapping, and continues on to possibly the best presentation of analytic continuation in the literature: Kodaira’s treatment of analytic continuation by integrals is particularly noteworthy.

Indeed, Kunihiko Kodaira’s *Complex Analysis* is a fantastic book — one of the best I have seen. As an analytic number theorist whose work intersects algebraic and differential geometry I will have occasion to consult it again and again in the future and I greatly look forward to the experience.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

1. Holomorphic functions; 2. Cauchy’s theorem; 3. Conformal mappings; 4. Analytic continuation; 5. Riemann’s mapping theorem; 6. Riemann surfaces; 7. The structure of Riemann surfaces; 8. Analytic functions on a closed Riemann surface.

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