**1. Entire functions (1954), by Ralph Philip Boas Jr.**

**1.1. Review by: A Pfluger.**

*Mathematical Reviews*MR0068627

**(16,914f)**.

The book gives a summary account of the extensive results obtained by mathematical research in the class of functions of exponential type in the last fifteen years. These functions are analytic and of exponential type in an angular space or in the plane (entire functions of exponential type). There is no complete or exclusive treatment of entire functions.

**1.2. Review by: Maurice Heins.**

*Science, New Series* **121** (3142) (1955), 390.

The theory of entire functions is a vast, rich chapter in the general theory of functions of a complex variable. Its literature is tremendous. The bibliography of the present monograph, which is concerned principally with the special class of entire functions of exponential type, consists of 17 pages. Intentionally, many topics of the general theory of entire functions, such as the Picard theorem and its developments, are omitted in this monograph. The result of this imposed restriction is a unity of theme and method. The interest of the special class of entire functions of exponential type is, of course, well known. As the author remarks, an account of the applications of functions of exponential type would call for a book by itself. What has been achieved here is a comprehensive account of the modern theory of functions of exponential type. The book is a valuable addition to the existing monographic literature on the theory of functions of a complex variable and is to be warmly recommended to both specialist and student.

**1.3. Review by: R W**.

*Science Progress* (1933-) **43** (171) (1955), 525.

Apart from polynomials and rational functions, the simplest kind of analytic functions are uniform functions with one essential singularity. When this is isolated and at infinity we get the class of integral functions. It is natural that the study of such functions should have been developed more completely than almost any other branch of function theory. What is remarkable is that this book on Entire Functions is the first book on integral functions to appear in English since the publication of Valiron's classical work. Professor Boas's book does not claim to be a comprehensive treatise on integral functions. It is largely concerned with the properties of functions of exponential type, that is, functions of at most order one, mean type, regular in the whole plane or in an angular region. This should have been made clear in the title since a number of important topics belonging to the general theory of integral functions have been omitted. ... Although this is a book for experts the introductory chapters are particularly good, and no one who is interested in integral functions should be without

**1.4. Review by: Archibald James Macintyre.**

*The Mathematical Gazette* **40** (331) (1956), 70-71.

This book deals with integral functions from a special point of view. Its aim is not unfairly described as the discussion of the validity of the classical interpolation expansions of Newton and Gauss, and their natural generalisations together with the immediate consequences of these expansions. ... The subject matter-apart from a 50 page introduction to the general theory of integral functions - is quite modern. There are very few references in the extensive bibliography outside the period 1940-1954. Nevertheless, the book is really an elementary one, the new ideas being notable for their originality of conception rather than their difficulty or depth. The author has managed to arrange in a systematic way a very large number of individual contributions (many being his own) and in considerable detail at the end of each chapter, an analysis of the relevant sources.

**1.5. Review by: J Korevaar.**

*Bull. Amer. Math. Soc.* **62** (1) (1956), 57-62.

This is the first book which is devoted primarily to the theory of analytic functions of exponential type, and it gives a practically complete account of the subject. ... Besides the theory of these functions the book contains a large chapter on entire functions in general, a chapter on the minimum modulus of an entire function, and a chapter on applications of functions of exponential type. The author and his publisher have rendered a great service by making so much material available in a single volume of moderate size and price. ... Besides the theory of these functions the book contains a large chapter on entire functions in general, a chapter on the minimum modulus of an entire function, and a chapter on applications of functions of exponential type. The author and his publisher have rendered a great service by making so much material available in a single volume of moderate size and price.

**2. Polynomial expansions of analytic functions (1958), by Ralph Philip Boas Jr and R Creighton Buck.**

**2.1. Review by: W F Newns.**

*Mathematical Reviews*MR0094466

**(20 #984)**.

The objects of this monograph are to explain a general method, the method of kernel expansion, of expanding analytic functions in series ... and to illustrate the power of the method by working out the details in a large number of important cases.

**2.2. Review by: Archibald James Macintyre.**

*The Mathematical Gazette* **43** (345) (1959), 235.

According to the preface the authors have tried to bring about a certain amount of order and completeness and to formulate results and methods in a fashion which will make them more generally accessible. They have succeeded in producing a very compact and readable account of a great deal of recent work and its more classical background. There is a twenty page introduction which reaches a form of generalised Appell polynomials. Proceeding to more special cases they discuss properties and expansion theories of the polynomials of Bernoulli, Laguerre, Hermite, Tchebycheff, Jacobi and many others. A brief but suggestive final chapter on applications to uniqueness problems and functional equations must be mentioned and the authors commended for their provision of a detailed contents list, index, bibliography, and list of special symbols.

**2.3. Review by: Casper Goffman.**

*Amer. Math. Monthly* **66** (8) (1959), 740.

This little book differs from most items in the Ergebnisse series in that, rather than being a survey of a field, it is a semi-expository presentation of the authors' approach to a topic. ... While the book is by no means light reading, it should be within the scope of a large segment of the mathematical public. In particular, it should make an excellent seminar topic.

**2.4. Review by: John Todd.**

*Mathematics of Computation* **15** (73) (1961), 104-105.

A great part of numerical analysis is concerned with polynomial approximation to analytic functions, and so this booklet appears of immediate interest to the numerical analyst. However, with numerical analysis in its present state, it is more relevant for studies in the general theory of functions of a complex variable or the theory of special functions. ... The material is accessible to those familiar with the classical methods of complex variable theory, and its study by numerical analysts is recommended. It will, for instance, encourage us to get off the real axis, reveal some thought-provoking "bad examples", and show us how mathematics should be written.

**3. A primer of real functions (1960), by Ralph Philip Boas Jr.**

**3.1. Review by: R L Jeffery.**

*Mathematical Reviews*MR0118779

**(22 #9550)**.

The author states his purpose in the following words: "My idea is to go reasonably far in a few directions with a minimum amount of special terminology." One finds that the special symbolism is restricted to those for inclusion, union and intersection. With this meagre equipment the author proves such theorems as the following. A non-empty closed set E in a complete metric space is also a complete metric space. A complete metric space is of the second category. The elements of the space C of continuous functions that have, even at one point, a finite derivative, even on one side, form a set of the first category in C. A continuous function is uniformly continuous on any compact set in its domain. Baire's theorem that a complete metric space is of the second category. ... The text is a valuable contribution to mathematical literature in that it sets forth in simple language and in short space the parts of the real variable theory that are essential to further study in the various fields of mathematics.

**3.2. Review by: T M Flett.**

*The Mathematical Gazette ***45** (354) (1961), 356-357.

There is a tendency for the modern mathematician to sit in his ivory tower and to concern himself only with the broad sweep of the view which lies within his gaze - he is rather like the landscape gardeners of the eighteenth century who considered anything less than a whole field as unworthy of their attention. It is good, therefore, to be reminded of the interest and the beauty which can be found in the more intensive cultivation of relatively small domains. In the book under review, we are given some of the results achieved by such an intensive cultivation, in that part of the field of "real variables" which deals particularly with the properties of continuity and differentiability. The book is not a systematic treatise, but is more in the nature of a course of informal lectures; the author has set out to tell readers who have little previous knowledge of the subject some of the results which he has himself found particularly interesting, and he has limited himself to the material that seemed essential for the results he had in mind, together with "as much related material as seemed interesting and not too complicated". His aim has been, too, "to preserve some of the sense of wonder that was associated with the subject in its early days but has now largely been lost". In this last aim Professor Boas has been strikingly successful, and his book presents a fascinating and very readable account of many of the interesting and beautiful results which lie in his chosen field. The sense of wonder is admirably preserved, and the book could well serve to revive the interest in analysis of those students to whom "analysis" at present too often means the proof, in successive annual stages of increasing rigour, of results learnt in outline at school.

**3.3. Review by: Fritz Herzog.**

*J. Amer. Statistical Association* **56** (293) (1961), 214-215.

Boas' monograph presents to the beginning student of the subject a good introduction into the concepts and methods usually presented in a course on Real Functions. The treatment is essentially the classical treatment of the subject, which in the reviewer's opinion enhances the value of the book from the expository standpoint. Topological concepts are kept to a minimum, while emphasis is put on those ideas which are at the basis of every rigorous treatment of real analysis. ... The exposition is usually clear and concise. More difficult parts of the book are printed in small type and can be skipped without harm. The book contains a good number of exercises, whose solutions are found at the end of the book. .... It might be doubtful whether, as the author claims, a student with a knowledge of calculus should be able to read this book. There is no doubt, however, that a slightly more advanced student (one who has had advanced calculus) will know a considerable amount of good mathematics after studying this primer.

**3.4. Review by: Pasquale Porcelli.**

*Amer. Math. Monthly* **68** (2) (1961), 192-193.

This excellent two-chapter little book treats a variety of specialised subjects in lively manner. The subject matter, for the most part, is selected from the foundations of analysis and ranges from important pathological examples to fundamental theorems and some of their applications. The first chapter presents some basic notions inherent to the foundations of analysis, and the last chapter treats some important properties of various classes of functions. Of particular interest here is the treatment of the linear function and convex functions. The entire presentation is (as the author states in his preface) informal and includes some sprinkled bits of wit and philosophy.

**3.5. Review by: I S Gál.**

*Bull. Amer. Math. Soc.* **68** (1) (1962), 10-12.

Long before this book was published Professor Boas told me of his plans to write a new Carus monograph on a less specialized topic than those presented in the last few volumes. He had already decided to write a first introduction to real variable theory and in order to have a guiding line he planned to include everything which is necessary to formulate and prove the following proposition: Suppose the continuous real valued function *f* of a real variable *x *has derivatives of all orders everywhere and for each *x *there is an order *n(x) *such that the *n(x)th *derivative of *f* vanishes at *x. *Then *f* is a polynomial function. ... The book is clearly written by a man who knows mathematics, has something to say and is able to communicate with others. Undergraduates and promising high school students could profit a great deal by reading it.

**3.6. Review by: Richard Bellman.**

*SIAM Review* **5** (3) (1963), 290.

This is an interesting and well-written introduction to real variable theory. It is highly recommended as a text for undergraduate and graduate courses and for the individual who wishes to forge his own path. Although the writing is leisurely (and a welcome relief from some of the modern pseudo-abstract expositions), a large amount of material is covered in an elegant fashion. A number of the results and proofs will delight even the expert. A very attractive feature is a large set of exercises, together with answers at the end of the book.

**4. A primer of real functions (3rd edition) (1982), by Ralph Philip Boas Jr.**

**4.1. From the Preface.**

In preparing this revision I have tried to resist the temptation to insert additional material; but a considerable number of references have been added to the notes.

**4.2. Review by: David Tall.**

*The Mathematical Gazette* **67** (442) (1983), 316.

This little book in an earlier edition has graced my shelf for many years, being as a source of personal inspiration and also for material used in university elegant and friendly in mathematical style claiming 'no previous knowledge assumed of the reader but he should have had at least a course in calculus'. It in 1960 and since then has grown in size from 190 to 232 pages, the latest additions considerable number of references added to the notes'. For those readers who have not come across the book in the last two decades, it concerns itself with sets and functions. ... I used it myself when learning about everywhere continuous nowhere differentiable functions which led to me writing a little article on the topic a couple of years ago in the 'Gazette'. I have reason to be thankful for its clear and helpful way of explaining abstruse ideas in analysis and have referred to it in my work on a number of occasions for nitty gritty results invaluable in tying up loose ends. A colleague of mine has used it as the basis of a more advanced analysis course. The fact that it has been reprinted and extended after over twenty years says all there is needed to say of this charming textbook.

**4.2. Review by: Nicholas D Kazarinoff.**

*Amer. Math. Monthly* **91** (3) (1984), 213-214.

Professor Boas' book is one that is both classical and modern in content, and in addition it excites the mathematical spirit - if it exists in the reader. Yet, beautiful and pedagogically sound as Professor Boas' book is, perhaps it is not the right book for the next generation. There is no mention of computing or numerical analysis either in it or in many widely used texts. An undergraduate course in real analysis (or in functions of several variables or advanced calculus) ought to serve the needs of the computer scientist and computer oriented mathematical scientist, as well as the future mathematician. For a basic analysis course introducing mathematical rigour and satisfying these needs, the concepts of number and function require reinterpretation with mention of the concrete and discrete and discussion of function evaluation algorithms. The constructive and algorithmic aspects of many other topics need to be taught. For example, the implicit function theorem could be followed by computation of Jacobians and Newton and/or quasi-Newton solution of systems of equations (algebraic or transcendental). Why should the cornerstone of the mathematics curriculum be relaid? The answer is that computer usage, from number crunching to investigation of geometric phenomena via computer graphics, is entering mathematics with strength and speed. Differential geometers, logicians, algebraists, and analysts of many varieties now use the computer. For those not entering mathematical research and teaching careers in universities, but who use mathematics in their work, the computer and computer graphics will be even more important. My preaching now gives way to praise of a respected colleague's work. Both the elderly and the young can enjoy Boas' 'Primer'. There are gems to delight any reader.

**5. A primer of real functions (4th edition) (1996), by Ralph Philip Boas Jr.**

**5.1. From the Preface.**

**5.2. Review by: Walter J Sanders.**

*The Mathematics Teacher* **90** (6) (1997), 499.

This 300-page fourth edition of Ralph Boas' Carus Monograph Number 13, first published in 1960 by the Mathematical Association of America, is an extension of the original by Harold Boas, son of Ralph Boas. This edition includes a new chapter on Lebesque integration. The Carus Monographs "are in tended for a wide circle of thoughtful people familiar with basic graduate mathematics ... who wish to extend their knowledge without prolonged study of journals." The reader should be aware that the reading can require effort, even for majors. ... I highly recommend this monograph to mathematics majors who wish to review or extend their knowledge of real analysis, and to workers in other fields who wish to learn of the major results of real analysis.

**6. Integrability theorems for trigonometric transforms (1967), by Ralph Philip Boas Jr.**

**6.1. Review by: R A Askey.**

*Mathematical Reviews*MR0219973

**(36 #3043).**

There has long been interest in the question of the relative sizes of a function and of its Fourier coefficients. Except for L^{2,}where there is Parseval's theorem, it was long ago realized that some auxiliary conditions must be placed on either the function or the coefficients before really satisfactory theorems could be obtained. ... This book is very well written and gives an up-to-the-minute survey; in addition, a number of open questions are given. One thing that these questions show is that we are still short of a number of special series which will provide counterexamples.

**7. Invitation to complex analysis (1987), by Ralph Philip Boas Jr.**

**7.1. From the Preface.**

This book is intended as a textbook for a first serious (senior or graduate level) course on complex analysis; but it is also intended to be useful for independent study. It assumes that its readers have had a course in advanced calculus or introductory real analysis (the material in the first part of my *A primer of real functions* [The Carus Mathematical Monographs, No. 13, Published by The Mathematical Association of America, 1960] is more than enough background).

**7.2. Review by: George Piranian.**

*Amer. Math. Monthly* **96** (4) (1989), 376-378.

Boas avoids the folly of an impossible definition by making a modest declaration. In his preliminary statement to students he writes: "Complex analysis was originally developed for its applications; however, the subject now has an independent and active life of its own, with many elegant and even surprising results." The declaration does not characterize complex analysis; but complex analysts know that no reasonable description of their territory could ever have remained satisfactory for more than a quarter century. An understanding of the term complex analysis requires at least peripheral participation. To those who by virtue of necessity or volition refrain from participation we might simply say that complex analysis began as the art of using complex-valued functions in the analysis of various physical problems and that today it is primarily the study, by analysis and synthesis, and with geometric, topological, algebraic, number-theoretic, or other cultural orientations, of complex-valued functions in spaces of one or more complex variables. ... Boas obviously believes in exercises, and he has confidence in the soundness of his pedagogical opinions. ... Boas is a master at conveying central truth without telling everything he knows. Although the book's title does not promise a sit-down dinner, that's what we get. An important aspect of the occasion is that no curriculum committee has caterer's rights. The MONTHLY'S editorial rules forbid recitations from the menu; but I can report that the host has chosen and prepared the food in his own inimitable way. The meal's nutritional balance is above reproach, and each dish presents itself attractively. The herbs come from Boas's private garden, and the beverages have aged in his cellar.

**7.3. Review by: Lawrence Zalcman.**

*Amer. Math. Monthly* **97** (3) (1990), 262-266.

Ralph Boas' 'Invitation to Complex Variables' is .. good ... The distillation of an intimate acquaintance with the field that extends over half a century, it is, in my opinion, the best undergraduate text ever written on the subject. The years of thought that must have gone into the making of this book are evident not simply in its organization and choice of topics (both of which are exemplary) but also in many little touches. Boas is the only author I know who approaches Morera's theorem from the proper direction (viz. Green's formula); and even when he comes to prove a result as familiar as the fundamental theorem of algebra, he has something interesting to show us. Again and again, one encounters illuminating discussions of matters ignored entirely by most other authors. Cases in point include the section on changing variables in improper integrals and the discussion of methods for calculating harmonic conjugates. Another nice feature of the book are the notes, which contain well-chosen references to the literature, attributions, pointed remarks, an occasional reminiscence, and some amusing speculations. As Boas observes early on, "complex analysis was originally developed for the sake of its applications." In addition to the usual applications to irrotational flows of incompressible, nonviscous fluids, there are discussions of the Plemelj formulas, Nyquist diagrams, and asymptotic series. But make no mistake about it: this is a mathematics book.

**7.4. Review by: Richard Rochberg.**

*American Scientist* **77** (4) (1989), 402.

This is a text for a first course in complex analysis. In addition to the standard topics based on the integral formulas and power series, there is an attractive collection of short discussions of additional topics including intuitive Riemann surfaces, non-Euclidean geometry, and Fourier series. This is a very friendly book. The style is informal, there are many figures, and there are lots of interesting asides - conceptual, historical, and whimsical. Boas's love of the subject comes through clearly and makes the book fun to read. If you are teaching an introductory course in function theory and do not use this book, at least tell your students about it. If you need to learn (or relearn) these things on your own, this book is one of the best sources.

**8. Lion hunting & other mathematical pursuits (1995), by Ralph Philip Boas Jr.**

**8.1. From the Introduction.**

Occasionally over the course of time, a person with the ability to do high quality research in mathematics also stands out in other ways, contributing to mathematics and the mathematical community in many areas. Ralph Philip Boas, Jr, was such a person: mathematician, author, editor, teacher, and administrator. This collection of his work represents an assortment of many of his lighter mathematical papers along with verse, stories, anecdotes, and recollections. In addition to highlighting his many different abilities, we hope that this volume will stimulate the reader in the way in which Ralph Philip Boas Jr intended to stimulate the readers of his articles when they were first published. The mathematics remains fresh and the comments on teaching are as cogent today as when he wrote them. The recollections and anecdotes included here, with a few exceptions, have not appeared in print before. These will provide the reader with a personal glimpse into the life of a mathematician who moved in the circles of the great mathematicians and scientists of his day, such as G H Hardy and many others. The stories have tantalizing depth and the verses offer imaginative and insightful views of many aspects of the life of a mathematician.

**8.2. Review by: Walter Sanders.**

*The Mathematics Teacher* **88** (7) (1995), 610.

If you are one who believes that all mathematicians are off in a world by themselves and not regular people - read this book. If you enjoy reading light, humorous vignettes about famous people - read this book. If you like the short, humorous poetry of Ogden Nash or the snappy observations of Dorothy Parker - read this book. If you just plain like mathematics - read this book! This tribute to Ralph Philip Boas, a fine mathematician, also shares with the reader a much broader profile of a man who contributed to the mathematics community as editor, teacher, administrator, and insightful humorist. Included are many of his lighter mathematical papers, along with verse, stories, anecdotes, and recollections of his interactions with most of the prominent mathematicians of 1930 to 1985. Perhaps the most important feature of this books is how it subtly makes the reader aware of the nature of mathematics.

**8.3. Review by: Leonard Gillman.**

*Amer. Math. Monthly* **103** (7) (1996), 607-609.

The stage is set by a twenty-page autobiographical sketch (taken from 'More Mathematical People', D J Albers et al., Harcourt Brace Jovanovich, 1990). It is a fascinating tour through Boas's school days, college years, and professional life, and includes a charming account of how he met his wife, Mary, as well as photographs of his family and of some of the eminent mathematicians he encountered. The book proper presents enough of a mix to delight readers of many stripes: reprints of a good number of Boas' expository mathematical papers (on Lion Hunting, Infinite Series, The Mean-Value Theorem and Indeterminate Forms, Complex Variables, Inverse Functions, Polynomials, The Teaching of Mathematics); several sections titled Recollections and Verse; Reminiscences by several close friends, students, and son Harold (also a mathematician); Reviews and other miscellany. What I found most engaging were the autobiography and other sections containing Boas' insightful comments about mathematics or about people and their foibles. His prose is simple and direct, and at the same time, elegant and witty ...