## Reviews of Lars Ahlfors' books

**1. Complex analysis. An introduction to the theory of analytic functions of one complex variable (1953), by Lars V Ahlfors.**

**1.1. Review by: Thomas Arthur Alan Broadbent.**

*The Mathematical Gazette*

**2**(38) (324) (1954), 149-150.

The English reader is well served with books on the complex variable, Edward Copson's volume being particularly noteworthy. This new work by Lars Ahlfors is worth adding to the teacher's shelf because of its re-distribution of emphasis, its variations in method, and some novelty of content. There are six chapters: Complex numbers; complex functions; complex integration; infinite sequences; the Dirichlet problem; multiple-valued functions. ... Altogether, a book to be warmly recommended as a clear, precise, and forward-looking introduction to the theory of functions of a complex variable. Having said this with, I hope, due emphasis, two small grumbles may be permitted. The author's uniform style sometimes masks the key result, and in a revision he might consider some re-wording so as to concentrate attention at the critical points. Secondly, some suggestions for further reading would be valuable.

**1.2. Review by: Pierre Lelong.**

Mathematical Reviews, MR0054016 **(14,857a)**.

The book has a presentation designed for first year students of the theory of analytic functions of a complex variable. The contents are classical; however, the book contains some very remarkable paragraphs that can serve as an introduction to further study and concern the Dirichlet problem, subharmonic functions, Riemann surfaces, and some other points. The talents of the author's style allows him to evoke very succinctly images that give the reader insight into and understanding of rigorous proofs; these qualities combine to make the book a very remarkable success in many ways.

**1.3. Review by: William Munger Boothby.**

*The Mathematics Teacher* **47** (2) (1954), 126.

Here is a book which is surely destined to become a standard text for a first-year graduate course in the theory of functions of a complex variable. The subject is treated with clarity and elegance and well illustrated with numerous problems. The emphasis throughout is on the geometric approach and power series are not introduced until the middle of the book. In addition to a thorough treatment of the standard material of such a course, some topics not usually found in books of this type are discussed, in particular a chapter on the Dirichlet Problem including a discussion of Perron's method.

**1.4. Review by: Pasquale Porcelli.**

*Mathematics Magazine* **27** (1) (1953), 47-48.

This is an excellent book that reflects the author's broad experience as both a contributor and a teacher of complex variable theory. The principle asset of the book is the spirit of integrity set forth by the author in the preface and rigidly adhered to throughout the text. ... The reviewer's overall estimate of the book can be best expressed by saying that he hopes it will become the accepted text in each class where complex variable theory is studied and where a book is used to outline the course.

**1.5. Review by: Pierce Waddell Ketchum.**

*Amer. Math. Monthly* **60** (10) (1953), 723-724.

Professor Ahlfors has fulfilled expectations that he would produce a scholarly and novel treatment of analytic function theory. The only disappointment which the reviewer experiences is that the author stopped too soon. One could wish for a continued discussion of more material in the same vein. It has been generally recognized that expositions of analytic functions in English have failed to make sufficient use of relevant topological tools. This is the more surprising in view of the great mutual influence that complex variable theory and topology have exerted upon each other. In the present book Ahlfors remedies this deficiency by making systematic use of topological techniques, and he has also provided brief but reasonably adequate introductions to the topological ideas which he uses. At the same time he has retained the flavour of classical function theory; and he has avoided the assembly-line format wherein every paragraph is labelled Theorem, Definition, Corollary, Remark, and the like.

**2. Complex analysis. An introduction to the theory of analytic functions of one complex variable (Second Edition) (1966), by Lars V Ahlfors.**

**2.1. Review by: Swarupchand Mohanial Shah.**

*Amer. Math. Monthly*

**75**(8) (1968), 924-925.

The first edition of this book appeared in 1953. The main changes in the second edition are the addition of a section on conformal mapping of polygons, a chapter on elliptic functions and a section on Picard's theorem on entire functions. ... The reviewer considers this book as one of the best on the subject. It is rigorous, readable and has a number of challenging exercises, some with hints. It is a suitable textbook for a two semester course on complex analysis for first-year graduate students.

**2.2. Review by: Pierre Lelong.**

*Mathematical Reviews*, MR0188405 **(32 #5844)**.

This new edition offers some additions, made possible, says the author in his preface, by the higher level of students entering university; accordingly, while still very basic in its first part, the book has been extended on important topics (topological notions, Riemann surfaces); it gives the reader an overview of modern methods (subharmonic functions, Dirichlet problem); a chapter on elliptic functions has been added and allows the author, using the modular function, to give the theorem of Picard. The work thus remains a remarkable introduction to a theory that retains an important place in basic teaching.

**2.3. Review by S L Green.**

*The Mathematical Gazette* **51** (378) (1967), 357-358.

First published in 1953, this is an admirable introduction to the theory of analytic functions of one complex variable. The author points out in the preface that there has been a marked change during the past decade in the quality of students entering American and other universities. They have been confronted with true mathematical reasoning at an earlier stage. This has led to some important changes from the first edition. The exponential and trigonometric functions are now defined by means of power series; the introduction to point set topology has been rewritten; normal families are more directly approached and the connection with compactness is emphasised; the Riemann mapping theorem has been combined with a section on the Schwarz-Christoffel transformation; a brief treatment of elliptic functions has been included; and exercise sections have been enlarged.

**3. Complex analysis. An introduction to the theory of analytic functions of one complex variable (Third Edition) (1978), by Lars V Ahlfors.**

**3.1. From the Preface:**

There are no radical innovations in the new edition. The author still believes strongly in a geometric approach to the basics, and for this reason the introductory chapters are virtually unchanged. In a few places, throughout the book, it was desirable to clarify certain points that experience has shown to have been a source of possible misunderstanding or difficulties. Misprints and minor errors that have come to my attention have been corrected.

**3.2. Review by: Harry Hochstadt.**

*SIAM Review* **22** (3) (1980), 378.

There is a classic story that when Newton's solution to the brachistochrone problem was forwarded anonymously to John Bernoulli, the latter's response was, "The lion is known by his paw." Likewise, in reading Ahlfors' Complex Analysis, one senses the sure touch of the master. in a mere 320 pages one finds far more material than can be taught in a one year graduate course (unless one wishes to offer "Complex Analysis for Masochists"). Much of the book is devoted to topics that are standard in an introductory course. There is a strong emphasis on the underlying geometric concepts and the author is also concerned with the computational skills required to carry out the necessary calculations. ... In summary, this book provides a thorough, extensive and sophisticated introduction to complex analysis. It is not an easy book and it is the exceptional graduate student who will be able to master its content without the assistance of an instructor. The first edition appeared in 1953 and there is every indication that this book will remain in print for many years to come. Such longevity is no accident. A good instructor using this textbook can give an exciting and challenging course on a topic with which everyone who wishes to be a mathematician must be familiar.

**4. Riemann surfaces (1960), by Lars V Ahlfors and Leo Sario.**

**4.1. Review by: H D Ursell.**

*The Mathematical Gazette*

**47**(359) (1963), 85.

Riemann surfaces were introduced into Mathematics in order to illuminate the nature of many-valued analytic functions of a complex variable, and in particular of algebraic functions. ... But Riemann surfaces occur also in other connections: and from a mere tool the concept has grown into a subject of study in its own right. Such it is here. Here a Riemann surface means an abstract "surface" with a "conformal structure": we are given the "angles of intersection" of curves drawn on the surface. ... A truly admirable work. But hardly for the beginner. *Formidable*, as the French say.

**4.2. Review by: M H Heins.**

*Mathematical Reviews*, MR0114911 **(22 #5729).**

A glance at the vast and very comprehensive bibliography of the book under review furnishes convincing evidence that there has been considerable activity in the study of the theory of Riemann surfaces since 1945 and that the theory has many different facets. Indeed, the output has been so great that it was necessary for great selectivity to be exercised in writing what must certainly be regarded as one of the authoritative accounts of the modern theory of Riemann surfaces. The choice of the material included and its disposition in the text depend upon many factors: the critical judgment of the authors, the importance of the material, and contemporary interest. Perhaps the spirit of the book can best be described by saying that it concentrates principally on the foundations essential for an understanding of the modern theory of Riemann surfaces and treats these meticulously. It is definitely intended that the treatise be accessible to students whose background may be limited. The student who has mastered the text will be very adequately prepared to pursue more special studies in the theory of Riemann surfaces.

**5. Lectures on quasiconformal mappings (1966), by Lars V Ahlfors.**

**5.1. Review by: P Caraman.**

*Mathematical Reviews*, MR0200442

**(34 #336).**

This is a new monograph on bi-dimensional quasi-conformal mappings ... The book contains several of the author's results; the style is clear; the proofs are simple; we find several examples and problems; but there is no index or references.

**6. Conformal invariants: topics in geometric function theory (1973), by Lars V Ahlfors.**

**6.1. Review by: Lipman Bers.**

*American Scientist*

**63**(3) (1975), 359-360.

This slim volume contains some of the deep and beautiful theorems in geo metric function theory that have been obtained during the past 50 years. The celebrated author is one of the creators of the theory, and several of the results are his own. ... The writing is lucid and incisive, and the elegant and complete proofs avoid any ponderous machinery. The author includes historical remarks and well-chosen exercises. All this in 150 pages! A masterpiece.

**6.2. Review by: M H Heins.**

*Mathematical Reviews*, MR0357743 **(50 #10211).**

*Conformal invariants: topics in geometric function theory* encompasses a wealth of material in a mere one hundred and fifty-one pages. Its purpose is to present an exposition of selected topics in the geometric theory of functions of one complex variable, which in the author's opinion should be known by all prospective workers in complex analysis. From a methodological point of view the approach of the book is dominated by the notion of conformal invariant and concomitantly by extremal considerations. Two courses of lectures of the author which were extant in note form, "Conformal mapping" given at Oklahoma A and M College 1951 and "Variational methods in function theory" given at Harvard University 1953, were employed in the preparation of the monograph under review. But, of course, the contents of the book reflect the considerable advances in geometric function theory since the appearance of the cited notes. The richness of the book will be apparent from the following summary of its contents. ... Even this condensed summary of the contents of *Conformal invariants* indicates its wide scope. It is a splendid offering.

**7. Möbius transformations in several dimensions (1981), by Lars V Ahlfors.**

**7.1. Review by: Gaven J Martin.**

*Mathematical Reviews*, MR0725161

**(84m:30028).**

The book is a well-written set of lecture notes from courses given at the University of Minnesota and the University of Michigan. The aim of the author is to give an elementary introduction to the geometric and analytic properties of Möbius transformations in n real dimensions and then to pursue some more advanced topics. ... In summary, this book is a timely arrival and fills a conspicuous gap in the current literature. It is virtually self-contained and provides a reference for many results confined to folklore. This book should be an essential addition to the library of anyone working in the general fields of quasiconformal analysis or discrete groups of hyperbolic isometries.

JOC/EFR November 2014

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