# Paul Dienes' books

We give below three books by Paul Dienes. For each we give extracts from some reviews of the book.

Leçons sur les singularités des fonctions analytiques (1913)

The Taylor series. An Introduction to the Theory of Functions of a Complex Variable (1931)

The logic of algebra (1938)

**Click on a link below to go to information on that book**Leçons sur les singularités des fonctions analytiques (1913)

The Taylor series. An Introduction to the Theory of Functions of a Complex Variable (1931)

The logic of algebra (1938)

**1. Leçons sur les singularités des fonctions analytiques (1913), by Paul Dienes.**

**1.1. Review by: Philip E B Jourdain.**

*The Mathematical Gazette*

**7**(107) (1913), 177.

These lectures were given at the University of Budapest, and the volume containing them forms part of the well-known collection of monographs on the theory of functions, edited by M Borel.

The systematic study of the singularities of analytic functions was begun by Hadamard, and his own results, together with those of Fabry, Leau, Le Roy, and others, were set forth in Hadamard's wonderful little book of 1901 on

*La série de Taylor*. The work of Borel and Mittag-Leffler on the representation of analytic functions opened the way for a general theory of singularities. and M Dienes gives a first sketch of such a theory in the present volume. His point of view is to make no restrictive hypothesis on the coefficients of the development, but to consider more or less particular singularities. He tries to represent the singularities by the nature of the divergence presented by the representation of the function at the point considered. The five chapters are devoted respectively to the researches of Hadamard and the order of a singular point, to a study of the singularities on the circle of convergence, to Borel's method of exponential summation, to the study of singularities by Mittag-Leffler's method, and to the general problem of singularities. The concise account of much of the recent work of Hadamard, Fatou, Fejér, Knopp, Schnee, Marcel Riesz, Borel, Hardy, and P and V Dienes makes this book a worthy successor of other volumes in the collection.

**1.2. Review by: N Nielsen.**

*Nyt tidsskrift for matematik*

**27**(AFDELING B) (1916), 67.

This publication, which is a volume in the "Collection of Monographs on the Theory of Functions, published under the direction of M Émile Borel," gives a short and clear presentation of: Hadamard's Studies, Study of Singularities on the Circle of Convergence, Study of Singularities by the Mittag-Leffler Method.

In particular, it must noted that the book's second and largest chapter on singularities in the circle of convergence is extremely well written, and that it can be recommended for a closer study by those who are interested in these theories.

**2. The Taylor series. An Introduction to the Theory of Functions of a Complex Variable (1931), by Paul Dienes.**

**2.1. Review by: Norman Miller.**

*The American Mathematical Monthly*

**39**(7) (1932), 418-420.

Professor Dienes has here performed a notable service to the mathematical world in assembling and ordering in a thoroughgoing manner the modern theories relating to Taylor series. From its title and subtitle one might suppose the book to be another elementary text book on complex function theory from the Weierstrass point of view. The title however is too modest. The book is a pioneer in its field and makes no inconsiderable demands on the maturity of its readers.

The first half of the book, after a foundation consisting of the theory of aggregates and fundamental ideas concerning functions of a real variable, presents most of the material of a first course in complex variable theory with the ideas of Cauchy and Weierstrass uppermost. The seven chapter headings of this part are I, Real Variables; II, Complex Algebra; III, Infinite Series; IV, Elementary Functions; V, Complex Differentiation; VI, Geometrical Language; VII, Complex Integration. The choice of material in these chapters is guided largely by the needs of the second part of the book. The treatment of specific functions (such as the doubly periodic functions) is off the main track of the author's purpose and is therefore dealt with briefly or not at all. On the other hand the geometrical ideas associated with Jordan's theorem, culminating in a treatment of Stieltjes integrals, are carried further than the requirements of a first course. Incidentally the cult of the analyst is shown in the author's use of 'mathematics' and 'geometry' as mutually exclusive terms.

The second half, which is the

*raison d'etre*of the book, has the following seven chapter headings: VIII, Biuniform Mapping, Picard's Theorem; IX, Representation of Analytic Functions; X, Singularities of Analytic Functions; XI, Overconvergence and Gap Theorems; XII, Divergent Series; XIII, The Taylor Series on its Circle of Convergence; XIV, Divergence and Singularities. The central problem as stated in the author's preface is that of detecting the properties of a function from the sequence of coefficients of its Taylor series and from the formal properties of these series. The pursuance of this central problem leads sometimes to theorems not always associated with Taylor series. Thus the inversion of Taylor series leads up to the classical theorems of Picard (on integral functions and essential singularities) and Riemann (on biuniform mapping). The detection and location of singularities by means of the Taylor coefficients and their relation to the character of the divergence occupies a large share of this half of the book. For this purpose a chapter is devoted to the theory of divergence and summability. This in the reviewer's opinion is not a treatment to which a beginner should come for his first ideas but rather a systematic exposition of the subject in a logical rather than a chronological order.

In regard to the manner of presentation, an effort is made to place due emphasis on important results so that the reader will not lose sight of the forest on account of the trees. This is very necessary in a book with so many laborious proofs which depend on obtaining finer and finer inequalities.

**2.2. Review by: G B Jeffrey.**

*The Mathematical Gazette*

**16**(221) (1932), 358.

This book is bound to exercise a considerable influence on the study of advanced mathematics in our modern Universities, for nothing quite on the same lines has hitherto appeared in English. The book seems to have been written with a dual purpose; it falls into two parts of about equal length. The first half of the book is occupied with the presentation of the elements of the theory of functions of a complex variable developed on the lines of Cauchy and Weierstrass. No doubt the author felt that the traditional lines of study of pure mathematics at the undergraduate stage are not well adapted as a preliminary to the study of modern developments of the subject. He therefore begins at the beginning and expounds "Real Variables", "Complex Algebra", "Infinite Series", "Elementary Functions". In a book of this kind it seems a little strange to treat the decimal representation of a number as fundamental, but this is only one example of several that could be quoted to show that this part of the book has been written with a didactic purpose, and that the author has his own views as to the way in which the student can best be helped to overcome the initial difficulties of this branch of study. Most of what is in this part of the book is already covered by existing English text-books and treatises, but no doubt many will find it a useful basis for the earlier part of an Honours course.

In the second and more valuable part of the book the purpose is to elucidate the problem of detecting the properties of a function from those of the coefficients in its Taylor expansion. Here the author has a clear field for, apart from Mandelbrojt's much smaller volume under the same title, no other effort has been made to give a systematic account of this work in English. Here we have a clear account of the elements of the work of Hadamard on singularities, of Mittag-Leffler on representation in the star-domain, of Ostrowski on over-convergence. The concluding chapters are devoted to a study of the behaviour of a series on its circle of convergence and to the relation of divergence of the series to singularities of the function.

The subject is one in which great advances have been made in recent time and in which a good deal of work is being done at present. The book is valuable introduction to a very interesting and live branch of Mathematics.

**2.3. Review by: H F.**

*Journal of the Institute of Actuaries (1886-1994)*

**63**(1) (1932), 110-111.

Taylor's Theorem is familiar to actuarial students both by its analogy to the simple formulae of interpolation and as a basis for series adaptable to approximate integration. It might be thought therefore that the book under review would be of more than passing interest to those members of the profession who still maintain their interest in mathematics. It is feared however that the non-specialist will find the mathematics of Mr Dienes' treatise too heavy for him, notwithstanding that, according to the preface, only the elements of ordinary calculus are supposed to be known by the reader (and are therefore deemed to be sufficient for him) before he embarks on a study of the book. The subtitle of the book is a clearer index to the subject-matter than the title itself, and actuarial students - in the wide sense of the term - might be dissuaded from a perusal of the book on learning that it is in effect an introduction to the Theory of Complex Variables.

To review the volume in detail is impossible in a limited space. The early part of the book deals with the elements of the Cauchy-Weierstrass theory of functions of a complex variable. The chapter on Real Variables is eminently satisfactory, as is the introduction to Complex Algebra in Chapter 11. The author's exposition of Jordan curves, while appearing rather obscure at a first reading, will repay closer study and will give the reader a good insight into a difficult conception. A discussion of the more familiar theorems of complex integrals completes this section of the book.

In the second part of the volume Mr Dienes has collated the results of recent researches in connection with the Taylor series. The author's knowledge of his subject is encyclopaedic, and not only has he succeeded in the task that he has undertaken in the collection of these researches, but he has contributed much original matter, especially in the realm of divergent series. The chapter dealing with this part of the subject can be heartily commended to the advanced student.

At one time it was the criticism of the mathematical writer that he rarely expressed himself succinctly and in good English. This reproach may have been deserved, but the modern school of mathematicians has shown conclusively that advanced treatises on abstruse subjects can be written with as much care for style and composition as an essay or a collection of

*belles-lettres*. A model for all writers on mathematical subjects is to be found, for example, in Whittaker and Watson's Modern Analysis. It is to be regretted that Mr Dienes has fallen from grace, for in certain places his enthusiasm for mathematics has obviously exceeded his care for his English. Such phrases as "We are going to ..." and "Same for" occur frequently: these modes of expression are uncouth and could easily be avoided. This, however, is a minor criticism.

The impression left after a careful perusal of the book is that Mr Dienes is to be congratulated on an important addition to the study of the Taylor series which is likely to be a standard work for the pure mathematician for many years to come.

**2.4. Review by: F E Relton.**

*Science Progress in the Twentieth Century (1919-1933)*

**27**(105) (1932), 154.

Dr Dienes has placed English-speaking students and mathematicians under a debt of gratitude by an ably executed and thoroughly painstaking piece of work. With commendable generosity he is almost lavish in his acknowledgments of the assistance rendered by others; but the work is obviously the outcome of years of assiduous devotion to a difficult task, and the result is altogether admirable. The author rightly states that no great mathematical armament is needed for the major portion of the book, and the treatment follows the lines of Cauchy and Weierstrass rather than Riemann; which is not to say that Riemann's ideas are ignored. There is a liberal supply of exercises for practice, and the author is wise in his mode of presentation. Too many writers present us with series of arguments to which nobody can refuse assent at any stage, only to leave us wondering at the end what it is all about. The sensation of letting a highly polished chain of reasoning slip through one's mind is pleasurable in itself. In the present work the author tells us quite explicitly in advance what it is he intends to prove. No one can pretend that the subject is easy reading; rather does the detailed study of it call almost for a special type of mind, even a special type of mathematical mind.

An efficient index and a fairly exhaustive list of errata are evidence of the care with which the work has been prepared, and when I add that there are no less than twenty-eight pages of bibliography, one sees that Dr Dienes is as efficient a cicerone as preceptor.

**2.5. Review by: Anon.**

*The Military Engineer*

**50**(335) (1958), 242.

An introduction to the theory of functions of a complex variable.

**2.6. Review by: Anon.**

*Nature*

**130**(3275) (1932), 188.

The first seven chapters of this book give the elementary properties of functions of a complex variable, ending with Jordan's theorem and a rigorous proof of Cauchy's theorem. These chapters are furnished with a variety of exercises. Chap, viii. discusses biuniform mapping and the theorems of Bloch, Schottky, Landau, and Picard. Chap. xi. deals with various means of representing a one-valued analytic function by an explicit formula. The problem of uniformisation is not discussed, as being beyond the scope of an introductory treatise, Chap. x, considers singularities and Chap. xi. overconvergence and gap theorems. Chap. xii. on divergent series gives a welcome and systematic discussion of generalised limits and sums. In Chap. xiii. this theory is applied to the Taylor series on its circle of convergence. Chap. xiv. discusses the relations between singularities and divergence. The whole book forms a very useful introduction to the theory of functions of a complex variable, and the author is to be congratulated on the manner in which he has systematised such an immense amount of material in a way which is calculated to give a proper perspective of the subject. The printing is good, but the numeration of the paragraphs is not sufficiently prominent for easy reference.

**2.7. Review by: Joseph Fels Ritt.**

This treatise conducts the reader from the elements of real variable theory into some of the furthest reaches of complex analysis. As we feel that the book will find its chief usefulness in connection with modern function theory, we shall describe first the more advanced chapters.

Chapter VIII deals with Picard's theorem and conformal mapping. By the simple method introduced in 1925 by Bloch, the theorems of Landau and Schottky and Picard's theorem are proved in succession. The Riemann mapping theorem for simply connected regions is proved by the most modern methods.

Chapter IX presents Weierstrass' factorization theorem for integral functions, Mittag-Leffler's theorem, Borel's method for the summation of divergent Taylor series and the representations of analytic functions in star domains, due to Mittag-Leffler and Painlevé.

Chapters X and XI deal with questions inspired, directly or indirectly, by Hadamard's dissertation. One finds here Hadamard's condition for the nonexistence of essential singularities on a circle of convergence and his theorem on the multiplication of singularities. Ostrowski's hyperconvergence is also presented.

Chapter XIII deals with conditions for a power series to represent a bounded function within its circle of convergence (Nevanlinna), with majorants (Hardy, Landau, Bohr), with convergence at singular points (Riesz),and with radial limits (Fatou).

In Chapter XII there is given an extensive treatment of the summation of divergent series. Applications are made in the final chapter (XIV), to the study of the types of divergence of power series at singular points. This last chapter also presents Hadamard's theory of the order of a singular point.

It would be difficult to overestimate the value, for advanced students, of these later chapters in Dienes' book. They reduce to didactic form a large section of the recent literature on complex analysis. These seven chapters, the second volume of Bieberbach's

*Funktionentheorie*and Montel's

*Fonctions Entières et Méromorphes*, give together quite a complete account of the more modern contributions to complex variable theory.

The first seven chapters of Dienes' book present what is commonly considered elementary complex variable theory, with the exception that, in Chapter V, one finds such topics as Schwarz's lemma, Hadamard's three-circle theorem, and Vitali's theorem. The elementary chapters contain a great amount of excellent material. One might mention the interesting treatment of hypercomplex numbers and also the valuable collections of exercises. However, we do not think it can be claimed that these chapters furnish a sound introduction to function theory. Let us consider, for instance, the question of analysis situs. Chapter VI contains a detailed proof of the Jordan separation theorem. On the other hand, the notion of sense receives no consideration, so that, in Chapter VII, one finds oneself integrating "with the area on the left," on a purely geometric basis, just as in older, frankly intuitive accounts of the subject. The treatment of the number system in Chapter I does not seem to us to be adequate for beginners. In Chapter III, integration is used on the basis of the student's knowledge of the calculus.

It thus appears that, although we have in recent years acquired excellent treatises on higher complex function theory, one must still refer students to the works of Osgood and of Tannery for thorough presentations of the elements of the subject.

Our comments on the earlier chapters of Professor Dienes' book are not to be interpreted as adverse criticism. We have sought merely to indicate what we consider the advantages of this treatise, which is the work of a distinguished authority and which will hold an important place in every mathematical library.

**3. The logic of algebra (1938), by Paul Dienes.**

**3.1. Review by: Saunders Mac Lane.**

*The Journal of Symbolic Logic*

**4**(2) (1939), 100.

This booklet gives a "realistic" slant to a mildly intuitionistic interpretation of real numbers and Boolean algebra. The realistic approach means that the integers, rational numbers, negative numbers, and finally real numbers are introduced successively in terms of their uses. The algebra of these numbers is then developed in elementary fashion, with a few "dialectic" arguments thrown in. Boolean algebra is considered as an algebra of "collections" of objects. The usual laws are obtained, except that the complement of a collection is not a collection, hence is introduced indirectly. This Boolean algebra is elaborately related to a new and useless classification of the moods of the Aristotelian syllogism. The consistency of this algebra is accepted on "realistic" grounds: "(i) the properties stated are actually possessed by collections of definite objects, (ii) consistency is perceived to be invariant under the devolving process of increasing ... the number of collection variables." The author gives a readable discussion of Brouwer's development of real numbers in terms of "evolving" decimal forms which successively specify more and more digits in the proposed decimal expansion and hence determine by degrees a more and more delimited "species" of real numbers. He nowhere mentions the more profound intuitionist position on existence theorems.

**3.2. Review by: L C Young.**

*The Mathematical Gazette*

**23**(256) (1939), 426.

Dr Dienes gives his personal "realist" conception of the notion of number without attempting any systematic criticism of other schools of thought.

**3.3. Review by: Barkley Rosser.**

*Bulletin of the American Mathematical Society*

**46**(1940), 15

In the preface Dienes says that it is his aim to clear the ground for a realistic discussion of the "crisis," and not to give a systematic description of the logical structure of algebra. To this end he has many discussions of the thought processes underlying our use of arithmetic and algebra. So long as he is discussing the more mathematical of these (counting, and the like), his treatment is sound, though antiquated. With some of the less mathematical thought processes, he is not so successful. In particular, his discussion of inference is extremely unsatisfactory.

Last Updated July 2022