The theory of functions of one complex variable is a privileged meeting place of many important branches of mathematics, and each one of them could claim ownership of this sparkling jewel. Complex analysis has always (and rightfully) had a privileged position in the education of a mathematics student. However, beyond the introductory undergraduate course on complex variables, with the usual manifold approaches, sometimes just emphasizing the computational aspects of the subject, many students and instructors, eschew or hide the all-important geometry underlying the whole theory, even in graduate courses. Some classical texts, such as Ahlfors’ Complex Analysis (McGraw-Hill, Third Edition, 1979) emphasize the geometrical aspects of the theory, yet others, such as Titchmarsh’s The Theory of Functions (Oxford, 1976) are more on the analytic side, but with a few exceptions, these books are no longer used a textbooks.
The book under review is a modern take on the subject, with the explicit goal of highlighting the geometric intuitions underlying the theory of functions. This is not an introductory textbook on complex analysis, whose elementary aspects are briskly covered in the first two chapters: From the definition of holomorphic and harmonic functions, complex integration, various versions of Cauchy’s theorem, singularities, residues and analytic continuation, to the Mittag-Leffler, Weierstrass and Runge theorems. The geometric approach is already visible even in these introductory chapters, which include the monodromy theorem and the Riemann Mapping theorem, and 19 figures illustrating several aspects of the theory being developed.
Chapter three is devoted to an in-depth study of the class of harmonic functions. Dirichlet’s boundary value problem on the unit disk is solved using the Poisson kernel, whose main properties are studied in the first two sections of this chapter, including the probabilistic aspects. The second part of this chapter is devoted to the study of the Hardy class of harmonic functions on the unit disk. Here the analytic requirements jump up to the level of Rudin’s Real and Complex Analysis (McGraw-Hill, Third Edition, 1987) for some basic results on functional analysis. This chapter also includes a section on entire functions of finite order, with proofs of Jensen’s formula and Hadamard’s factorization theorem. To help develop geometric intuition, the last section of this chapter displays eight conformal plots generated by mapping rectangular grids to the complex plane.
In accordance with the geometric emphasis taken by the author, the remaining two thirds of the book take the theory of functions of one complex variable to its natural continuation, so to speak, introducing the notion and main properties of the natural domain of these functions: Riemann surfaces. Thus, the geometric heart of the book starts in Chapter four, where Riemann surfaces are introduced as 2-dimensional connected Hausdorff spaces with a countable basis for its topology and provided with a maximal atlas of conformal structures. Many examples illustrate this concept, from open domains in the complex plane and the Riemann sphere to complex projective algebraic curves.
After introducing the natural maps between Riemann surfaces and the notion of degree of a map, we find the basic Riemann-Hurwitz theorem relating the genus of the (compact) Riemann surfaces involved, the degree of the map and the total branching number. It is perhaps important to point out that the genus of a compact Riemann surface is introduced in the first section of the Appendix, along with several other topological notions and constructions, such as covering spaces, the fundamental group, and the homology and cohomology of surfaces. Included in this chapter are sections devoted to study Riemann surfaces defined as quotients of actions by a discontinuous group and groups of Möbius transformations.
One section is devoted to study the field of meromorphic functions on an elliptic curve and the relation of these constructions to the classical theory of elliptic integrals. Even in this introductory chapter, the important case of compact Riemann surfaces is emphasized. This case will be taken again in Chapter seven, where the theorems of Riemann-Roch, Abel and Jacobi will be proved in complete detail. Meanwhile, Chapter five gives a second approach to the introduction of Riemann surfaces, retaking the problem of analytic continuation. Thus, Riemann surfaces appear as the unramified or ramified surfaces associated to an analytic germ. One highlight of this chapter is the construction of the complex projective algebraic curve associated to a compact Riemann surface. This construction is illustrated with several examples, including the case of elliptic curves.
The next major result, the Hodge decomposition theorem, is the subject of Chapter six, where differential forms, their integrals and residues are discussed. Here is also where we find the important result on the existence of nonconstant meromorphic functions on a Riemann surface, as a consequence of Hodge’s theorem. The last chapter of the book is devoted to the classification theorem of Riemann surfaces, that is, the uniformization theorem, first in the simply-connected case with detailed arguments, and then in the non simply-connected one, in a rather sketchy way. The last section of Chapter eight is an introduction to the study of Fuchsian groups and their associated fundamental regions.
The geometric emphasis of the author is visible throughout the whole book, with many figures that illustrate key points and ideas. Every chapter comes with a list of problems to give the prospective student a chance to prove her/his understanding of the subject. At the end of every chapter there are some notes giving some historical context of the material just treated or pointing the reader to the literature for further reading.
The book could be used for self-study or in a graduate course: Like many others, the book started as a set of lecture notes used by the author in his courses, so it has already been tested as such. Recent additions to the literature that have some intersections with the book under review or go in the same general directions are Varolin’s Riemann Surfaces by Way of Complex Analytic Geometry (AMS, 2011) or the massive two-volume Complex Analysis 1, by Freitag and Busam (Springer, 2005) and Complex Analysis 2, by Freitag (Springer 2011).
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is email@example.com.