Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable

Citation

Ahlfors, L. V. (2021). Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable (3rd ed.). American Mathematical Society (AMS Chelsea Publishing, vol. 385).

Chapter Summary

The AMS Chelsea volume is a 331-page reprint of the third edition, which was first published in 1978. It has eight chapters, a preface and an index. The publisher describes the book as a rigorous introduction, on an elementary level, to analytic functions of one complex variable, intended for first-year graduate students and advanced undergraduates. The presentation is slanted toward the geometric approach, with much material on conformal mappings, the Riemann mapping theorem, Dirichlet's problem, the monodromy theorem and the regions for which the Cauchy integral theorem holds; it also covers power series, contour integrals and infinite products, and the treatment of special functions is concise but reasonably complete. In the preface to the third edition Ahlfors writes that he still believes strongly in a geometric approach to the basics, so the introductory chapters are virtually unchanged from earlier editions.

Chapter 1: Complex Numbers

The opening chapter, as in the six-chapter first edition, is devoted to complex numbers.

Chapter 2: Complex Functions

The second chapter turns to functions of a complex variable. S. L. Green noted that from the second edition the exponential and trigonometric functions are defined by means of power series.

Chapter 3: Analytic Functions as Mappings

This chapter treats analytic functions geometrically, as mappings of regions. W. M. Boothby's early review notes the emphasis on the geometric approach, with power series not introduced until the middle of the book.

Chapter 4: Complex Integration

Complex integration develops the Cauchy integral theorem and its consequences. The publisher stresses the attention given to the kinds of regions for which the Cauchy theorem holds, and a reviewer of the first edition commended the systematic use of topological ideas with brief introductions to what is needed.

Chapter 5: Series and Product Developments

The fifth chapter covers series expansions and infinite products, topics the publisher lists among the analytic parts of the book, together with the concise treatment of special functions.

Chapter 6: Conformal Mapping. Dirichlet's Problem

This is the chapter that carries the geometric core: conformal mapping, the Riemann mapping theorem and the Dirichlet problem, the existence of a harmonic function with given boundary values. The first edition's chapter on the Dirichlet problem included Perron's method, and Green notes that the Riemann mapping theorem is combined with a section on the Schwarz-Christoffel transformation.

Chapter 7: Elliptic Functions

Elliptic functions were added as a chapter in the second edition. According to Pierre Lelong's review in Mathematical Reviews, the modular function is used there to give Picard's theorem.

Chapter 8: Global Analytic Functions

The last chapter deals with analytic functions in the large, and the publisher names the monodromy theorem among the subjects of the book's global theory.

Key Concepts

Chapters 1–5: Foundations

• Analytic functions as mappings: Chapter 3, reflecting the book's geometric emphasis.
• Cauchy integral theorem: Chapter 4, and the question of the regions for which it holds.
• Power series, contour integrals and infinite products: Chapters 4–5.

Chapters 6–8: Geometric function theory and global theory

• Conformal mapping and the Riemann mapping theorem: Chapter 6.
• Dirichlet's problem: Chapter 6, the existence of a harmonic function with prescribed boundary values.
• Elliptic functions: Chapter 7.
• Monodromy theorem and global analytic functions: Chapter 8.

Critical Analysis

Strengths

• Geometric slant. The publisher describes the book as slanted toward the geometric approach, and Ahlfors states the same conviction in his preface. Pierce Ketchum, reviewing the first edition in the American Mathematical Monthly, praised its systematic use of topological techniques together with brief introductions to the topological ideas needed, while keeping the flavour of classical function theory.
• Concision and clarity. AMS describes the presentation as concise, clear and thorough, and still fresh more than thirty years after the last revision. Harry Hochstadt, in SIAM Review (1980), found far more material in about 320 pages than can be taught in a one-year graduate course, and described a thorough, extensive and sophisticated introduction.
• Exercises and range. S. M. Shah, reviewing the second edition in the American Mathematical Monthly, called it rigorous and readable with a number of challenging exercises, some with hints, suitable for a two-semester course for first-year graduate students.
• Topics beyond the standard course. Boothby noted a chapter on the Dirichlet problem, including Perron's method, as a topic not usually found in books of this type, and Lelong's review of the second edition mentions subharmonic functions, Riemann surfaces and elliptic functions.

Limitations

• Not easy. Hochstadt wrote that this is not an easy book and that it is the exceptional graduate student who will master it without the assistance of an instructor. Broadbent, in an early review, said that the uniform style sometimes masks the key result.
• Reading suggestions. The same early review of the first edition regretted the lack of suggestions for further reading.
• More than one course. Hochstadt remarks that the book holds far more than can be taught in a one-year graduate course, so any class has to select, and readers working alone need to decide what to leave out.
• A reprint. The 2021 volume reprints the third edition, so its material is that of the late 1970s.

Where It Fits in the Course

This is the concise, geometric graduate text in the complex analysis list. For a gentler start, Beck, Marchesi, Pixton and Sabalka's A First Course in Complex Analysis (free and openly licensed, aimed at a one-semester undergraduate course) and Brown and Churchill's Complex Variables and Applications build the theory with applications in view. Stewart and Tall's textbook develops the rigorous theory for undergraduates while keeping geometric ideas in sight, and Tristan Needham's Visual Complex Analysis gives the pictures and physical intuition behind the same results, which pairs well with Ahlfors's geometric chapters. Stein and Shakarchi's volume is the other rigorous graduate-level option, with more emphasis on connections to Fourier analysis and number theory;

How to Use This Book

The book suits first-year graduate students and strong advanced undergraduates with a solid background in real analysis and some point-set topology; the text itself begins with the complex numbers.

• Core: Chapters 1–6 give the standard graduate course, with Chapter 6 as the main geometric destination.
• Optional or second term: Chapter 7 (elliptic functions) and Chapter 8 (global analytic functions) for a second semester or independent study.
• How to read: work with a pencil, treat the exercises as part of the text, and consult a more computational book alongside; Needham or Brown and Churchill can fill gaps in intuition or technique.