1.1. Review by: Thomas John l'Anson Bromwich.
The Mathematical Gazette 3 (57) (1906), 316-317.

This tract contains a number of striking and important theorems on indefinite integrals, which are due to Liouville very largely. ... Nearly half the tract is devoted to the integrals of algebraical functions ...

2.1. From the Preface.

This tract has been long out of print, and there is still some demand for it. I did not publish a second edition before, because I intended to incorporate its contents in a larger treatise on the subject which I had arranged to write in collaboration with Dr Bromwich. Four or five years have passed, and it seems very doubtful whether either of us will ever find the time to carry out our intention. I have therefore decided to republish the tract.

The new edition differs from the first in one important point only. In the first edition I reproduced a proof of Abel's which Mr J E Littlewood afterwards discovered to be invalid. The correction of this error has led me to rewrite a few sections (pp. 36-41 of the present edition) completely. The proof which I give now is due to Mr H T J Norton. I am also indebted to Mr Norton, and to Mr S Pollard, for many other criticisms of a less important character.

2.2. Review by: Philip E B Jourdain.
Science Progress (1916-1919) 11 (43) (1917), 518.

In the first edition (1905) of this useful tract the subject of indefinite integration of what the author called and calls "elementary functions" was given in a very thorough and systematic form. An "elementary function" is a member of the class of functions which comprises rational functions, algebraic functions - explicit or implicit, the exponential function, the logarithmic function, and all functions which can be defined by means of any finite combination of the symbols proper to the preceding four classes of functions. The integration of all such functions was treated with particular reference to the work of Abel, Liouville, and Tschebyschef; and an admirable Bibliography of the papers by these and other mathematicians was given.

The new edition differs from the first only in one very important point. Mr J E Littlewood discovered that a proof of Abel's given in the first edition was invalid; a new proof due to Mr H T J Norton is given in this edition, and a few sections on pp. 36-41 (cf. pp. 66-67) are consequently completely rewritten. There are many other minor alterations in this edition, and the Bibliography has been expanded.

This is probably the most useful Tract in the whole series for one who means to study mathematics systematically.

3.1. Review by: Henry Ricardo.
Mathematical Association of America (22 April 2006).
https://www.maa.org/press/maa-reviews/the-integration-of-functions-of-a-single-variable

G H Hardy (1877-1947) was one of the most respected and influential analysts/number theorists of the first half of the twentieth century. The volume under review is a hardcover reprint of his 1916 classic tract. I use Hardy's own word "tract" because this is not a textbook: There are no exercises and no applications (of course!).

In six elegant and precisely written chapters (Charles Dickens rather than Dan Brown), Hardy leads the reader through various classes of functions (see the Table of Contents), focusing on finding antiderivatives - or, as Hardy puts it, solving the equation $\large\frac{dy}{dx}\normalsize  = f(x)$. In analysing such an equation, Hardy is interested in "the functional form of $y$ when $f(x)$ is a function of some stated form." He does not discuss the theory of integration (Riemann sums and so forth). Chapter VI contains a nice discussion of Liouville's results on integration in finite terms. The Bibliography, consisting of references in French, German, and English, is history preserved in amber.

This is a book to have in the library as a reference; but its usefulness on one's personal bookshelf has been superseded by many research and expository journal articles and by computer-oriented books such as Handbook of Integration by D Zwillinger and the recent Irresistible Integrals by Boros and Moll.

4.1. From the Preface.

The ideas of Du Bois-Reymond's Infinitärcalcül are of great and growing importance in all branches of the theory of functions. With the particular system of notation that he invented, it is, no doubt, quite possible to dispense; but it can hardly be denied that the notation is exceedingly useful, being clear, concise, and expressive in a very high degree. In any case Du Bois-Reymond was a mathematician of such power and originality that it would be a great pity if so much of his best work were allowed to be forgotten.

There is, in Du Bois-Reymond's original memoirs, a good deal that would not be accepted as conclusive by modern analysts. He is also at times exceedingly obscure; his work would beyond doubt have attracted much more attention had it not been for the somewhat repugnant garb in which he was unfortunately wont to clothe his most valuable ideas. I have therefore attempted, in the following pages, to bring the Infinitärcalcül up to date, stating explicitly and proving carefully a number of general theorems the truth of which Du Bois-Reymond seems to have tacitly assumed - I may instance in particular the theorem of iii. § 2.

I have to thank Messrs J E Littlewood and G N Watson for their kindness in reading the proof-sheets, and Mr J Jackson for the numerical results contained in Appendix III.

5.1. Review by: Eric Harold Neville.
The Mathematical Gazette 12 (179) (1925), 514-515.

The missionary work of this tract was accomplished by the first edition. Everyone is familiar now with >- and 0, and there is even risk in the disseminated knowledge that higher mathematics uses the very language which comes naturally to the schoolboy whose ideas on tendencies and limits are still amorphous; no one can acquire a knack of using epsilons in a manner both plausible and worthless, but teachers will have to be alert to know when a verbal argument corresponds to a real grasp of the subject. This is not an imaginary danger. In his last book, after disavowing any intention of introducing double limits, Dr Leathem offers a proof of an approximation to the sine which depends on the assertion that if each term in a convergent sequence of functions of a variable is of order higher than the third, the limit must be of order higher than the third; this assertion is disproved by such an example as nx4/(1 + nx), and it is incredible that if the argument had been conducted in symbols a mathematician of experience would have supposed that it did not involve a double limit or that it was valid.

Save that the proof that any function finitely definable by means of logarithms and exponentials is ultimately continuous and monotonic has been recast completely, the text of the seven chapters of the original tract is reproduced with comparatively little modification in the first five chapters of this edition, which contain therefore an account of scales of infinity in general and of logarithmico-exponential scales in particular.

6.1. From the Preface.

The publication of this tract has been delayed by a variety of causes, and I am now compelled to issue it without Dr Riesz's help in the final correction of the proofs. This has at any rate one advantage, that it gives me the opportunity of saying how conscious I am that whatever value it possesses is mainly due to his contribution to it, and in particular to the fact that it contains the first systematic account of his beautiful theory of summation of series by 'typical means'.

The task of condensing any account of so extensive a theory into the compass of one of these tracts has proved an exceedingly difficult one. Many important theorems are stated without proof, and many details re left to the reader. I believe, however, that our account is full enough to serve as a guide to other mathematicians researching in this and allied subjects. Such readers will be familiar with Landau's Handbuch der Lehre von der Verteilung der Primzahlen, and will hardly need to be told how much we, in common with all other investigators in this field, owe to the writings and to the personal encouragement of its author.

6.2. Review by: Philip E B Jourdain.
Science Progress in the Twentieth Century (1906-1916) 10 (39) (1916), 500-501.

This extremely able tract is, owing to the war, issued without Dr Riesz's help in the final correction of the proofs. This has given Mr Hardy the opportunity of mentioning in the preface the value of Dr Riesz's contributions to the book and the whole theory in general. Further, Mr Hardy refers to his great debt to the writings and personal encouragement of Dr Edmond Landau, and this partly explains the very touching dedication of this little volume. The subject of Dirichlet's series and their applications to the theory of numbers up to 1909 has been very fully dealt with by Landau in his Handbuch der Lehre von der Verteilung der Primzahlen of the above date.

6.3. Review by: Anon.
The Mathematical Gazette 8 (124) (1916), 307-308.

Dirichlet's series - or rather a special kind of them - were first introduced into analysis by Dirichlet, primarily with a view to applications in the theory of numbers. Somewhat generalised, they may be defined as infinite series of terms $a_{n} exp (\lambda_{s}, s)$, where the $\lambda$s are real increasing numbers and s is a complex variable. The first theorems involving complex values of $s$ were due to Jensen in 1884 and 1888; and the first attempt to construct a systematic theory of functions defined by Dirichlet's series was made by Cahen in 1894 in a manner which, in spite of or possibly because of errors, stimulated most of the later researches in the subject. The theory of the functions in question presents very curious differences from the theory of analytic functions, which may, of course, be studied from the point of view of being defined by series of powers. In the first place, the region of convergence of a Dirichlet's series is a semi-infinite plane, and in the second place, a Dirichlet's series convergent in a portion of the plane only may represent a function regular in a wider region of it or even all over it, and the result is that many of the peculiar difficulties which attend the study of power series on the circle of convergence are extended, in the case of Dirichlet's series, to wide regions of the plane or even the whole of it (pp. 9-10). After giving the elementary theory of convergence, the authors deal with the formula for the sum of the coefficients of a Dirichlet's series; the summation of divergent Dirichlet's series by "typical means" (of which this is the first systematic account, and which is chiefly due to Dr Riesz); theorems on the summability of these series; further developments of the theory of functions represented by them ; and the multiplication of Dirichlet's series (which has specially important applications in the analytical theory of numbers). There is, finally, a very full bibliography which supplements that given up to 1909 in Handbuch der Lehre von der Verteilung der Primzahlen.

The work of Mr Hardy, among others, has brought, of recent theory of Dirichlet's series into the foreground of interest, and competent presentation of the subject will help to create and more interest.

6.4. Review by: George Ballard Mathews.
Nature 96 (1915), 312.

One observation in this tract is liable to misunderstanding. It may be true that Cahen is the first to discuss f(s) systematically as a function of a complex variable s; but the first step in this direction was taken by Riemann in his famous paper on the distribution of primes. To arithmeticians, at any rate, it is the development of Riemann's results that is of principal interest at present; other problems of the same kind naturally present themselves, as, for example, the frequency of cases where $p$ and $(p + 2)$ are both primes, like (5, 7) or (11, 13). The new analysis may throw some light on these and other dark places. In any case, this tract will be welcome for its concise statement of known facts, and its bibliography, which supplements that of Landau.

7.1. From the Publisher.

This classic work explains the theory and formulas behind Dirichlet's series and offers the first systematic account of Riesz's theory of the summation of series by typical means. Its authors rank among the most distinguished mathematicians of the twentieth century: G H Hardy is famous for his achievements in number theory and mathematical analysis, and Marcel Riesz's interests ranged from functional analysis to partial differential equations, mathematical physics, number theory, and algebra.

Following an introduction, the authors proceed to a discussion of the elementary theory of the convergence of Dirichlet's series, followed by a look at the formula for the sum of the coefficients of a Dirichlet's series in terms of the order of the function represented by the series. They continue with an examination of the summation of series by typical means and of general arithmetic theorems concerning typical means. After a survey of Abelian and Tauberian theorems and of further developments of the theory of functions represented by Dirichlet's series, the text concludes with an exploration of the multiplication of Dirichlet's series.

8.1. From the Preface.

This book was planned and begun in 1929. Our original intention was that it should be one of the Cambridge Tracts, but it soon became plain that a tract would be much too short for our purpose.

Our objects in writing the book are explained sufficiently in the introductory chapter, but we add a note here about history and bibliography. Historical and bibliographical questions are particularly troublesome in a subject like this, which has applications in every part of mathematics but has never been developed systematically.

It is often really difficult to trace the origin of a familiar inequality. It is quite likely to occur first as an auxiliary proposition, often without explicit statement, in a memoir on geometry or astronomy; it may have been rediscovered, many years later, by half a dozen different authors; and no accessible statement of it may be quite complete. We have almost always found, even with the most famous inequalities, that we have a little new to add.

We have done our best to be accurate and have given all references we can, but we have never undertaken systematic bibliographical research. We follow the common practice, when a particular inequality is habitually associated with a particular mathematician's name; we speak of the inequalities of Schwarz, Hölder, and Jensen, though all these inequalities can be traced further back; and we do not enumerate explicitly all the minor additions which are necessary for absolute completeness.

We have received a great deal of assistance from friends. Messrs G A Bliss, L S Bosanquet, R Courant, B Jessen, V Levin, R Rado, I Schur, L C Young, and A Zygmund have all helped us with criticisms or original contributions. Dr Bosanquet, Dr Jessen, and Prof Zygmund have read the proofs, and corrected many inaccuracies. In particular, Chapter III has been very largely rewritten as a result of Dr Jessen's suggestions. We hope that the book may now be reasonably free from error, in spite of the mass of detail which it contains.

Dr Levin composed the bibliography. This contains all the books and memoirs which are referred to in the text, directly or by implication, but does not go beyond them.

8.2. Review by: Børge Jessen.
Matematisk Tidsskrift. B (1935), 44-45

Already in 1928, Prof Hardy, in a London Mathematical Society Lecture entitled "Prolegomena to a chapter on inequalities", spoke of the author's intention to write a book on inequalities. In this at once entertaining and thought provoking lecture (which is printed in the Journal of the London Mathematical Society, vol. 4) he reproduces, among other things, a note, apparently by Harald Bohr, that "all analysts spend half their time hunting through the literature for inequalities which they want to use and cannot prove," and later, after discussing his intention to write the book, he adds: "And I am sure that we will deserve the thanks of the mathematical world, even if we do not do it particularly well." Now, then, with such great anticipation, the book has arrived, and let it be said that it satisfies all one's hopes. The book spans a wide register, both in terms of methods and results.

8.3. Review by: Edward C Titchmarsh.
The Mathematical Gazette 18 (231) (1934), 341-344.

A treatise by three of the leading mathematicians of the day on a subject of which we all learnt something at school should attract the notice of every mathematician. There are treatises on almost every conceivable mathematical subject, but apparently never before has one been written simply on inequalities. Yet there are many more or less simple inequalities which, as the authors say, are "of daily use" in analysis. This book was originally planned as one of the series of Cambridge Tracts, but the authors soon found that they needed more scope than a tract would give them. The book contains ten chapters, of which the first is introductory and explanatory. Chapters 2-6 contain a thorough and systematic discussion of the inequalities "in daily use", and centre round the theorem of the arithmetic and geometric means, Hölder's inequality, and Minkowski's inequality. The last four chapters contain an account of more recent work in which the authors themselves have played a large part. Altogether there are 405 theorems, each embodying an inequality or a statement immediately connected with inequalities.

8.4. Review by: Gilbert A Bliss.
Science, New Series 81 (2110) (1935), 565-566.

This book is devoted to a systematic and critical study of a number of inequalities which are fundamental in mathematical analysis, and to the presentation of a wealth of others which the authors have encountered in their wide experience, many of which have been subjects of their own investigations. It is unique in its field for many reasons, but especially on account of the great variety of results presented and the thoroughness with which the inequalities have been analysed and generalised. Mathematical investigators will find it an indispensable source of information.

Most mathematicians regard inequalities as auxiliary in character and would perhaps not think of them as constituting a domain of principal interest apart from applications. In reading the book it is a pleasant surprise, therefore, to find that the theory of inequalities is a fascinating subject in itself, and to see how effectively the theory may be systematised and correlated by skilful analysts. The authors have achieved much in this regard, and the results of their efforts indicate the possibility of still further interesting correlations in the future. Their plan is outlined in excellent fashion, with regard to both content and method, in Chapter I, which concludes with some helpful advice to the reader who may be interested in principal results rather than details. ...

No description of the book here reviewed would be complete without mention of the very valuable lists of theorems and examples at the ends of the chapters. If proofs were given for all these results the book would be expanded beyond reason, but in most cases the necessary arguments are clearly indicated or references are cited. This is only one of many features which insure the great value of the book as a contribution to our modern mathematical literature.

9.1. From the Preface.

This book is a development of two lectures delivered at the Harvard Tercentenary Conference of Arts and Sciences in the fall of 1936. The first of these was published in Vol. 44 of the American Mathematical Monthly, and is reprinted here, as Lecture I, without change. The second has expanded gradually until it fills the rest of the book.

I have given many lectures on Ramanujan's work since 1936, isolated lectures to a number of universities and societies in America and England, and connected courses in Princeton and Cambridge. Lectures II-XII contain most of the substance of these courses, with the rearrangements and additions required to fit them for publication. In this sense they are genuine lectures, and they are written throughout in a lecturer's style.

The contents of the book are described quite accurately by its title. It is not a systematic account of Ramanujan's work (though most of his more important discoveries are mentioned somewhere), but a series of essays suggested by it. In each essay I have taken some part of his work as my text, and have said what occurred to me about its relations to that of earlier and later writers. But even when I digress furthest, when I am writing, for example, about Rademacher's work in Lecture VIII, or about Rankin's in Lecture XI, 'Ramanujan' is the thread which holds the whole together.

Dr R A Rankin has read the whole of the book both in manuscript and in proof, and has made a very large number of important suggestions and corrections. I have also to thank Dr W N Bailey, who helped me to revise Lecture VII; Mr F M Goodspeed, who read and criticised several lectures, and in particular Lecture XI; and Prof G N Watson, without whoso aid I could hardly have written Lecture XII. The photograph of Ramanujan was given me by Dr S Chandrasekhar, formerly Fellow of Trinity College: I regret that I cannot state the name of the actual photographer. A considerable part of the bibliography was compiled for me by Dr V Levin. But my first thanks are due to Harvard University, to whose invitation the book owes its existence.

9.2. Review by: Derrick H Lehmer.
Science, New Series 93 (2426) (1941), 620-621.

In 1936, as part of the Harvard Tercentary celebration, the author delivered two lectures on the subject: The Indian Mathematician Ramanujan and his work. The first lecture, much the more biographical of the two, was published in volume 44 of the American Mathematical Monthly, and is reprinted in the present book as Lecture I. The original second lecture, which dealt more with Ramanujan's contributions to mathematics, has now been expanded to make the other eleven lectures. Most of this expansion, naturally enough, is the result of having adequate space more than the bare facts about Ramanujan's work. Some of it, however, is due to the fact that since 1936 much new work has been done on certain of his problems. Lectures VIII and X on the asymptotic theory of partitions and on Ramanujan's function ƒ(n) contain a great deal of recent mathematics. The connections between Ramanujan's work and that of other writers, especially those who came after him, are traced in each lecture with many interesting comments. Each lecture closes with a. set of valuable notes, and there is a bibliography of over 100 papers on problems suggested by Ramanujan.

Ramanujan surpassed all other men in his ability to produce striking formulas. Opening the book at almost any place one is sure to find some arresting equality, something typographically complicated with bizarre exponents and coefficients. In this respect Ramanujan differed from the typical modern mathematician who is trying to find the simplest possible relationships between extremely general concepts. The actual source of one of these startling formulas is often mysterious, and the author frequently indulges in interesting speculations. It is true that Ramanujan discovered a number of interesting results experimentally, but his experiments were never on a large scale. Many of the formulas which he gave as approximately correct are not as accurate as he had surmised, as a little experimenting would have shown. ...

The reader who is unable to appreciate Ramanujan's fields of endeavour, but who is interested in what constitutes and causes mathematical genius will find the book absorbing. There are several interesting passages on the subject of "proof" and what it meant to Ramanujan. The book is written in the best Hardy style.

9.3. Review by: Muhammad R Siddiqi.
Current Science 10 (7) (1941), 343-344.

"Prof Hardy, as he rightly claims, is the greatest authority on Ramanujan, and any book by him dealing with the life and work of our illustrious compatriot is bound to be an event of unusual importance in mathematical circles. Prof Hardy had written a memoir on Ramanujan, which was published along with the latter's collected papers. But it was the general belief that this was not enough from one who saw the departed savant and talked with him almost every day for several years, and who stood in a unique relation to "the most romantic figure in the recent history of mathematics". We are glad to see that Prof Hardy has tried to supply this long-felt want in the book under review. It originated in two lectures given at Harvard on the occasion of its tercentenary conference in 1936. Since then Prof Hardy has given many lectures on Ramanujan's work to a number of Universities in England and America, and also regular courses at Princeton and Cambridge.

The first lecture deals with Ramanujan's life and career. Prof Hardy reaffirms with considerable conviction his earlier opinion of Ramanujan's genius; but he also with draws one or two statements made in the previous memoir which he now considers quite ridiculous sentimentalism. For instance, Prof Hardy wrote earlier: "He (Ramanujan) would probably have been a greater mathematician if he could have been caught and tamed a little in his youth; he would have discovered more that was new, and that no doubt of greater importance. On the other hand he would have been less of a Ramanujan and more of a European Professor, and the loss might have been greater than the gain." Prof Hardy refutes this last sentence most strongly, saying that there was no gain at all when the college at Kumbakonam rejected the one great man they had ever possessed, and that the loss was irreparable. This leads him incidentally to pronounce a scathing criticism of our inefficient and inelastic educational system which can fail to recognise the genius of a Ramanujan.

As Prof Hardy aptly points out, Ramanujan "had been carrying an impossible handicap, a poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe". It should be admitted, however, that in spite of this severe handicap, Ramanujan did beat the accumulated wisdom of Europe in several instances. He was by far the greatest formalist of his time, and one of the three great formalists of all time, the other two being Euler and Jacobi. Prof Hardy is right when he says: "There have been a good many more important, and I suppose one must say greater, mathematicians than Ramanujan during the last fifty years, but no one who could stand up to him on his own ground. Playing the game of which he knew the rules, he could give any mathematician in the world fifteen."

There are some very good passages in the book which give us a true insight into Ramanujan's character. In one such pas sage, Prof Hardy depicts him as a man who, when he was living in Cambridge in good health and comfortable surroundings, was in spite of his oddities, 'as reasonable, as sane and in his Way as shrewd a person as any one here'. The picture that the author wants to present to us is "that of a man who had his peculiarities like other distinguished men, but a man in whose society one could take pleasure, with whom one could drink tea and discuss politics or mathematics, the picture in short, not of a wonder from the East or an inspired idiot, or a psychological freak, but of a rational human being who happened to be a great mathematician". Some people may think that having been through a very trying life in the early days, and having achieved world fame almost overnight, Ramanujan might have become somewhat of an egoist. Prof Hardy dispels this doubt completely by saying that "he (Ramanujan) was not particularly interested in his own history or psychology; he was a mathematician anxious to get on with the job". ...

We are strongly of the opinion that a copy of this book must be in every mathematical library as well as in the hands of everyone who wishes to acquaint himself with the working of the mind of an inspired mathematician who, like Abel and Galois, was snatched away in his prime, and but for whose premature death many chapters in mathematics would have been enriched beyond measure.

10.1. Review by: James R Newman.
Scientific American 203 (1) (1960), 188.

Hardy's book consists of his 12 lectures on the life and work of the extraordinary Indian prodigy Srinivasa Ramanujan, a most romantic and paradoxical figure and in some respects a very great mathematician.

10.2. Review by: Paul T Bateman.
Pi Mu Epsilon Journal 3 (2) (1960), 72-73.

"I have set myself a task in these lectures which is genuinely difficult and which, if I were determined to begin by making every excuse for failure, I might represent as almost impossible. I have to form myself, as I have never really formed before, and to try to help you to form, some sort of reasoned estimate of the most romantic figure in the recent history of mathematics; a man whose career seems full of paradoxes and contradictions, who defies almost all the canons by which we are accustomed to judge one an other, and about whom all of us will probably agree in one judgment only, that he was in some sense a very great mathematician." - From the first lecture.

The celebrated Indian mathematician Srinivasa Ramanujan was born in December, 1887. He had no university education, and worked unaided in India until he was twenty-seven. In 1913, however, letters from Ramanujan to the late G H Hardy in England gave unmistakable evidence of his powers, and he was brought to Trinity College, Cambridge, in April, 1914. There he had several years of very fruitful research activity in the analytic theory of numbers, much of it in collaboration with Hardy. He was ill (of tuberculosis) from May, 1917, onwards, returned to India in February, 1919, and died in April, 1920.

"Ramanujan was, in a way, my discovery. I did not invent him - like other great men, he invented himself - but I was the first really competent person who had the chance to see some of his work, and I can still remember with satisfaction that I could recognise at once what a treasure I had found .... My association with him is the one romantic incident in my life." - From the first lecture.

Hardy was obviously proud of his role in the Ramanujan story, and he had every right to be. Great as Ramanujan's gifts were, it has to be admitted that without Hardy's interest, encouragement, assistance, teaching, and collaboration, Ramanujan's name might well be comparatively unknown today. And not only did Hardy help Ramanujan directly while the latter was alive, but after Ramanujan's death Hardy did much to spread Ramanujan's reputation and further the appreciation of his work.

"The contents of the book are described quite accurately by its title. It is not a systematic account of Ramanujan's work (though most of his more important discoveries are mentioned somewhere), but a series of essays suggested by it. In each essay I have taken some part of his work as my text, and have said what occurred to me about its relations to that of earlier and later writers. But even when I digress furthest, when I am writing, for example, about Rademacher's work in Lecture VIII, or about Rankin's in Lecture XI, 'Ramanujan' is the thread which holds the whole together." - From the preface.

This book is probably the outstanding example of an exposition of the work of one great mathematician by another. Hardy's writing skill is justly famous, and he applies it admirably to the task of explaining Ramanujan's mathematical triumphs - and also his failures. While there is much hard mathematics in this book when the author gets down to cases and while very few readers can expect to follow all the discussions in detail, nonetheless there is plenty that is accessible to the less expert reader. On the whole the book makes fascinating reading for anyone with some interest either in the theory of numbers or in the recent history of mathematics.

11.1. From the Publisher.

Ramanujan occupies a unique place in analytic number theory: his formulas, identities and calculations are still amazing mathematicians three-quarters of a century after his death. Many of his discoveries seem to have appeared as if from the ether. His mentor and primary collaborator was G H Hardy. Here, Hardy collects 12 of his own lectures on topics stemming from Ramanujan's life and work. The topics include: partitions, hypergeometric series, Ramanujan's tau-function and round numbers.

11.2. From the Preface (by Bruce C Berndt).

Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, by G H Hardy, was originally published by Cambridge University Press in 1940. It was reprinted by Chelsea in 1960, and again, with corrections, in 1978. The book has been out of print for several years.

Since Hardy's lectures on Ramanujan were published in 1940, the subject of each of the twelve chapters has grown considerably. In this reprinting, we provide commentaries on each chapter to bring readers up to date on some of the important activity that has occurred since 1940. Our comments on certain chapters are brief, because their contents are mostly found in Ramanujan's published papers, for which extensive commentary is provided in the American Mathematical Society's new reprint of Ramanujan's Collected Papers.

11.3. Review by: Steve Abbott.
The Mathematical Gazette 85 (504) (2001), 541-542.

The story of Srinivasa Ramanujan's extraordinary life is now well known. This state of affairs doubtless owes a great deal to G H Hardy, who wrote this book following a series of lectures he gave in the late thirties. It was originally published by Cambridge University Press in 1940 and later reprinted by Chelsea in the sixties. This new AMS Chelsea version is supplemented with a commentary by Bruce C Berndt, the leading authority on Ramanujan's work. Berdt has made an extensive study of Ramanujan's papers and is able to shed light in some areas where Hardy's information was incomplete.

Eleven of the 'twelve lectures' are an artifice in that they represent a distillation of several of Hardy's lectures and lecture courses from the late thirties. However, they are presented in the same 'lecture format' as the first chapter, 'The Indian mathematician Ramanujan' a genuine lecture from the 1936 Harvard Tercentenary Conference of Arts and Sciences.

The first lecture is primarily biographical, detailing Ramanujan's life and Hardy's reaction to meeting and working with him. The other chapters are more mathematical, giving accounts of several areas of mathematics where Ramanujan made a contribution. Because Hardy worked with Ramanujan a great deal, there are many biographical glimpses in these lectures and several attempts by Hardy to understand how Ramanujan came to make such exceptional discoveries. The topics covered include descriptions of probabilistic number theory, the Hardy-Littlewood circle method, and Ramanujan's epic work with hypergeometric functions and integration.

It was fortunate that Ramanujan, whose notebooks were famous for obscuring his methods through their lack of proof, should work with Hardy, an accomplished communicator. In this book we are treated to an masterly exposition of some difficult mathematics, leavened with the biographical material. As Hardy explains in the preface, the title is accurate: the book is not a full account of Ramanujan's work, but Ramanujan is the connecting thread that holds it all together.

12.1. From the Preface.

This tract is based on lectures which each of us has given in Cambridge or elsewhere. There are already a good many books on the subject; but we think that there is still room for one which is written in a modern spirit, concise enough to be  included in this series, yet full enough to serve as an introduction to Zygmund's standard treatise.

We have not written for physicists or for beginners, but for mathematicians interested primarily in the theory and with a certain foundation of knowledge In particular we assume acquaintance with the elements of Lebesgue's theory of integration: it is impossible to understand the theory of Fourier series properly without it, and experience shows that it is well within the powers of any clever  undergraduate. The actual knowledge needed here can be acquired quite easily from chapters x - xii of Titchmarsh's Theory of Functions. As regards the theory of trigonometrical series, the book is 'officially' complete in itself; but we recognise unofficially that practically all our readers will have some knowledge of the subject (such as the substance of Titchmarsh's chapter xiii) already.

We have naturally been forced to omit much which we should have liked to include.  In particular we have no space for the inequalities of Young and Hausdorff, Marcel Riesz's theorem concerning conjugate series, theorems concerning Cesaro summation of general order, or uniqueness theorems involving summable series. And we give no results about special series except a few which we require to illustrate the general theory.

The notes at the end are not systematic; we have  inserted only such references and comments as we could make shortly and seemed to us likely to be useful. In particular we make no attempt to give any adequate idea of the history of the subject: Euler, Fourier himself, Poisson and Dirichlet are hardly mentioned. It is quite impossible in an account like this to do any justice to the great mathematicians who founded the theory.

12.2. Review by: J H Pearce.
Nature 154 (1944), 130-131.

This important new Cambridge tract is concerned with the modern developments of the mathematical theory of Fourier series. The character of this theory was radically altered during the decade 1900-10 by three fundamental discoveries: (1) the Lebesgue integral (1902-6); (2) the Fejér theorem (1904); (3) the Riesz-Fischer theorem (1907).

12.3. Review by: Edward C Titchmarsh.
The Mathematical Gazette 28 (281) (1944), 164.

Considering the comparative completeness of the subject, and the long time during which it has been familiar, the literature of Fourier series is still not very extensive. There are chapters on it in many books, but until recently a complete account had hardly been attempted. The modern theory, depending on Lebesgue's theory of the integral, was introduced in Lebesgue's Leçons sur les Series Trigonometriques. This was published as long ago as 1906, and was unfortunately never re-written. The most complete and up-to-date account is Zygmund's treatise in the series of Polish monographs, published in 1935. This is rather a formidable book for anyone but the expert. It is of very close texture, and there is much more in it than its 320 pages might lead one to expect.

For all these reasons, the appearance of a Cambridge tract by Hardy and Rogosinski is to be welcomed. It is definitely a book for the pure mathematician, but it should appeal to a large class of readers. It is really quite easy to read. A fair background of knowledge is an advantage, though very little is assumed. The theory is introduced in a "modern" manner; "Hilbert space", "strong convergence", "closure", "completeness", and general methods of summability are the conspicuous features. This is in the nature of the subject. Problems such as those depending on "mean" or "strong" convergence have a complete solution, and rightly take their place in the forefront of the theory. The classical convergence problem falls into its proper place, though, when we come to it, it is treated fully, in some ways more completely than in any other book. There appears to be no finality about this problem. The last word on the subject at present is a refinement, due to Gergen, of a theorem originally given by Lebesgue. This was published in 1930, and is Theorem 58 of this tract.

The book concludes with a very interesting series of notes and references. We believe that it will give a great many mathematicians all over the world as much pleasure to read, as it must have given the authors pleasure to write. We can imagine the satisfaction with which the English author, at any rate, must have retired undefeated, having filled his hundred pages and proved his century of theorems.

13.1. Review by: Michael Berg.
Mathematical Association of America (16 July 2013).
https://www.maa.org/press/maa-reviews/fourier-series

There's light in the darkness ... in the form of the time-honoured maxim pronounced by Niels Hendrik Abel: study the masters, not their pupils. Admittedly one should temper this judgment somewhat: acquiring the expertise and acumen required for studying a master entails an apprenticeship which involves plenty of preliminary sources. But every so often one has the great good fortune of being able to follow a preliminary course mapped out by a master: after all, even Chopin wrote many, many études.

And so did G H Hardy. In fact, early on in his career, and early in the 20th century, Hardy set himself the task of bringing British pure mathematics to parity with its continental counterpart, the standard having been set in the preceding two centuries largely by Germany and France. And so there flowed from Hardy's fountain pen a number of fabulous introductory and expository works - just think of his unsurpassed A Course of Pure Mathematics and, at a higher level, one of his magna opera, An Introduction to the theory of numbers, written with E M Wright. Well, here, in the book under review, we encounter another entry in this list: the composition Hardy wrote together with W W Rogosinski on Fourier Series.

It's a very short book, but dense and rich (like a diamond, one might well say). Say the authors:

"There are already a good many books on the subject [of course, nothing like what we're faced with now (ed.)]; but we think that there is still room for one written in the modern spirit [bis.], concise enough to be included in this [Cambridge tract] series, yet full enough to serve as an introduction to Zygmund's standard treatise ... We have not written for physicists or for beginners, but for mathematicians interested primarily in the theory and with a certain foundation of knowledge." They go on to say that the latter "can be acquired quite easily from chapters x - xii of Titchmarsh's Theory of Functions."

Isn't this fabulous?: Titchmarsh to Hardy-Rogosinski to Zygmund - a well-lit path to true hard analysis with a classical flavour. (By the way, not all that long ago, I had the honour of reviewing Zygmund's authoritative tome in this very column.)

Finally, when we dive into the text itself, we proceed from Fourier series in Hilbert space, through treatments of convergence and summability, to "general trigonometrical series," swiftly and elegantly. It's all scholarship of the highest order, of course, written in what is easily recognised as Hardy's unsurpassed style, and cannot be praised too highly. God bless Dover Publications for re-issuing this book.

14.1. From the Preface (by J E Littlewood).

Hardy in his thirties held the view that the late years of a mathematician's life were spent most profitably in writing books; I remember a particular conversation about this, and though we never spoke of the matter again, it remained an understanding. The level below his best at which a man is prepared to go on working at full stretch is a matter of temperament; Hardy made his decision, and while of course he continued to publish papers, his last years were mostly devoted to books; whatever has been lost, mathematical literature has greatly gained. All his books gave him some degree of pleasure, but this one, his last, was his favourite. When embarking on it he told me that he believed in its value (as he well might), and also that he looked forward to the task with enthusiasm He had actually given lectures on the subject at intervals ever since his return to Cambridge in 1931, and had at one time or another lectured on everything in the book except Chapter XIII.

The title holds curious echoes of the past, and of Hardy's past. Abel wrote in 1828: 'Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever. ' In the ensuing period of critical revision they were simply rejected. Then came a time when it was found that something after all could be done about them. This is now a matter of course, but in the early years of the century the subject, while in no way mystical or unrigorous, was regarded as sensational, and about the present title, now colourless, there hung an aroma of paradox and audacity.

14.2. Review by: Otto Szász.
Bulletin of the American Mathematical Society 56 (5) (1950), 472-473.

Hardy died on December 1, 1947; during his lifetime the theory of divergent series and its applications developed into an important branch of modern analysis. It is only natural that the book bears all the marks of his own research work, but it is also a comprehensive presentation of this vastly expanded subject. The final galley proofs were read and completed by some of his younger collaborators, in particular L S Bosanquet. ... The large amount of material evidently did not permit more detail concerning applications to trigonometric series, or a discussion of the subject of orthogonal series and of multiple series. It seems worthwhile to write another book on these subjects.

14.3. Review by: A C O.
Science Progress (1933-) 38 (151) (1950), 534.

This book is the last which Prof Hardy wrote. It is on a subject on which he frequently lectured in Cambridge, and many of us who attended those lectures look back on them as some of the finest he gave. They have now been preserved in a form in which they retain all their freshness, and can be appreciated by a much wider audience.

The book deals with the theory of divergent series in the form which has now become classical. The first two chapters contain many historical references which everyone will find of great interest. Chap. Ill deals with the general theory of linear transformations by Toeplitz matrices. The results of this chapter are applied in the following chapters, where special methods of summation are considered.

Chaps. V and VI are on Arithmetic Means, and the first of these gives what is probably the most complete account in the literature of the Cesàro, Hölder and Riesz means.

Tauberian theorems for Arithmetic means and for Power series are considered in Chaps. VI and VII respectively. Here Hardy is writing of work to which he, together with Littlewood, made very great contributions. Indeed one cannot help recalling a remark of Wiener's that, had not usage established otherwise, Tauberian theorems might have been more appropriately named Hardy -Littlewood theorems. The chapters on Tauberian theorems are probably the finest in the book. The closely related work of Fuchs and Rogosinski on inclusion theorems for Hausdorff means is given in Chap. XI, and Chap. XII contains a remarkably lucid account of Wiener's Tauberian theorems.

Mathematical literature has been greatly enriched by this work. We are also very much indebted to Dr L S Bosanquet, who assisted Hardy throughout, and saw the remaining proofs through the press, when Hardy was unable to continue working.

14.4. Review by: J H Pearce.
The Mathematical Gazette 33 (305) (1949), 217-218.

This book by the late Professor Hardy contains the matter of his courses of lectures delivered in Cambridge at intervals between 1931 and his death in 1947, revised and enlarged in a form suitable for publication. The subject is one in which, for many years, he had displayed an absorbing interest, and his many important contributions to its development, both alone and in collaboration with Littlewood, have rightly led pure mathematicians in all parts of the world to regard him as its undisputed master. As all his friends and pupils well knew, Hardy possessed a rare gift, the ability of a great mind to convey some of his greatness to others less gifted than himself, by means of exceptionally lucid exposition, both orally and in writing. In the most intricate mathematical analysis he was able to guide his audience through the difficulties in a manner that made them appear almost disarmingly simple, so much so that many a student has confessed to having attended the same course of lectures twice, for the sheer joy of watching once again his master's superb technique and expository skill.

It is no exaggeration to say that the theory of 'Divergent Series' illustrates modern analysis in one of its most intricate (and at the same time one of its most attractive) forms, and anyone who has occasion to read or merely to refer to this book will at once see the force of the remarks in the preceding paragraph. Few subjects have had a more "chequered career" than this one, and many mathematicians, from Leibniz onwards, have succumbed to the temptation to fall under its spell. Euler seems to have been the first mathematician to make any systematic use of divergent series, from which by formal manipulation he was able to produce many correct and important results. When, however, as was inevitable from time to time, he arrived at a conclusion, either partially incorrect or paradoxical, he frequently took refuge in metaphysics. A little later Poisson and Fourier continued their study: indeed Poisson's writings contain, in effect, the definition of Abel summability of a trigonometrical series. As the development of pure mathematics became more orthodox, however, along the lines followed by Cauchy and Abel, series which did not converge were gradually banished, and not until late in the nineteenth century did they reappear.

14.5. Review by: Harry R Pitt.
Nature 167 (4254) (1951), 743-744.

While it is true, as Prof J E Littlewood has remarked in the preface to this book, that the subject of divergent series has lost the sensational and paradoxical flavour which it had when he and Prof G H Hardy began their work on Tauberian theorems, it still remains one of the most important branches of modern analysis and certainly one of the most attractive.

The peculiar formal elegance of the subject and the interest of the delicate techniques which it requires are brought out to the full in this book, and it is hard to imagine a more pleasant or readable presentation of a subject of comparable difficulty. While it is plainly impossible to cover the whole field of divergent series exhaustively in a single work, the book contains enough to give a non-specialist reader a balanced picture of the main development of the theory into its modern form and, at the same time, to provide the specialist with an invaluable reference book from which it is only a short step to matters of contemporary research interest.

A short historical survey of the many ideas which have been held about divergent series is followed by an account of the general theory of regular transformations of series, including the Toeplitz-Schur theorem on necessary and sufficient conditions for regularity, Knopp's theorem on positive transformations, and a discussion of the extension of the theory to integral transformations of functions of a real variable.

The main part of the book is then devoted to a full account of most of the standard methods of summation and the Abelian and Tauberian theorems associated with them. It includes proofs of the classical Tauberian theorems of Hardy and Littlewood for Cesaro and Abel summation, with a sketch or the original proof of the latter by the repeated differentiation technique, as well as the later and simpler proof of Karamata. The basic ideas in these
earlier works are discussed in more detail in their application to the Hardy- Littlewood theorem for Borel summation with the $O(n^{-1/2})$ type of Tauberian condition.

The book also contains a large amount of material on the less-familiar summation methods. In particular, there is a very attractive introduction to the theory of Hausdorff methods, including Euler's, and an account of the elegant formal relationship between the latter and Borel summation.

The part of the book given to Wiener's methods is short, but it is enough to provide an account of the general Tauberian theorem in a form which makes
its application to all the standard methods seem natural and straightforward. The power and generality of the method is illustrated by its use in the 'one-sided' form of the Borel-Tauberian theorem.

Among the many other topics discussed are the Euler-Maclaurin summation formula, the multiplication of series and the Fourier kernels of summation matrices; and there are many illuminating comments and suggestions in the notes at the end of each chapter.

15.1. Review by: Allen Stenger.
Mathematical Association of America (15 December 2014).
https://www.maa.org/press/maa-reviews/divergent-series

This book is primarily about summability, that is, various methods to assign a useful value to a divergent series, usually by forming some kind of mean of the partial summands. It has only a little bit about asymptotic series, that is, divergent series for which it is possible to obtain a good approximation to the desired value by truncating the series at a well-chosen term. The present book is an AMS Chelsea 1991 reprint of the 1949 Oxford University Press edition. (There were several corrected reprints by Oxford, running through 1973, but this reprint seems to be the original 1949 edition.) This was Hardy's last book, as he died at the end of 1947 after the book was written and while it was still working its way through the press; several younger colleagues saw it through the proofreading process.

The summability material is primarily about Tauberian theorems, an area that was developed in the early 1900s by Hardy and J E Littlewood. Tauberian theorems say that if a series is summable by a particular method and satisfies an additional hypothesis (usually that it is not too wiggly) then it in fact converges, or in some cases that it is summable by another method. These theorems are important in prime number theory and other fields.

There is also material here on summability for analytic continuation and for multiplication of series, the Euler-Maclaurin summation formula, and a good exposition of Wiener's general Tauberian theorem (which, despite the name, belongs more to Fourier analysis than to summability).

The treatment is classical, and omits more modern material such as functional analysis methods and the algebraic view of Wiener's general Tauberian theorem. The treatment is specific rather than general, and each summability method is handled separately without much attempt to put the methods in a general framework. The book is a reference more than a textbook, as there are no exercises. Very Good Features: several detailed indices, and extensive bibliographic notes at the end of each chapter.

Although the material is still valuable, there are better books today.