# G N Watson's books

Neville Watson was the single author of two books and a joint author with E T Whittaker of a third. We note that 'Whittaker and Watson', as their famous text has been known, came about as a second edition of the book

*A Course of Modern Analysis*(1902) with E T Whittaker as the single author. The fourth edition of this work was published in 1927 and reprinted in 1935, 1940, 1946, 1950, 1952, 1958, and 1962. Watson's Bessel functions book ran to two edition, the second, published in 1944, was reprinted in 1952, 1958, 1962, 1966. We present some extracts from reviews and prefaces below. We list the different editions of these books separately below.

**1. Complex integration and Cauchy's theorem (1914), by G N Watson.**

**1.1. From the Preface.**

My object in writing this Tract was to collect into a single volume those propositions which are employed in the course of a rigorous proof of Cauchy's theorem, together with a brief account of some of the applications of he theorem to the evaluation of definite integrals.

My endeavour has been to place the whole theory on a definitely arithmetical basis without appealing to geometrical intuitions. With that end in view, it seemed necessary to include an account of various propositions of Analysis Situs, on which depends the proof of the theorem in its most general course of a memoir by Ames; my indebtedness to it and to the textbooks on Analysis by Goursat and by de la Vallée Poussin will be obvious to those acquainted with those works.

I must express my gratitude to Mr Hardy for his valuable criticisms and advice; my thanks are also due to Mr Littlewood and to Mr H Townshend, B.A., Scholar of Trinity College, for their kindness in reading the proofs.

**2. A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (2nd edition 1915), by E T Whittaker and G N Watson.**

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

*Mind*

**25**(100) (1916), 525-533.

There is a great contrast between the first and second editions of Prof Whittaker's work: in the second edition Mr Watson's collaboration has apparently affected not only the second half of the book, but also the fundamental and logical aspects. The second half in the second edition is quite the best of its kind in the English language; but we are mainly concerned here with the first half of the book, which concerns, according to the first edition 'those methods and processes of higher mathematical analysis which seem to be of greatest importance at the present time'. It is in this part that the second edition shows traces of the enormously important modern work on exact treatment of the subject-matter of analysis. It is not possible to say with precision whether any particular alteration is due to Prof Whittaker or to Mr Watson, but it is possible to make more or less plausible guesses based on a reading of Mr Watson's very useful Cambridge Tract on

*Complex Integration and Cauchy's Theorem*(Cambridge University Press, 1914). ...

Whereas in the first edition, complex numbers were hurried in by the words: 'To express the result of this and similar operations [the extraction of the square root of a negative number], we make use of a new number, denoted by the letter i; this is defined as a quantity which' satisfies certain laws. Of course it is obvious that this is no attempt even at an 'introduction' in the authors' sense. In the second edition, however, complex numbers are defined in quite the fashionable way as pairs of real numbers, and a, very modern spirit is shown by the fact that equality is not redefined for complex numbers. But it is rather unsatisfactory to read that 'we are at liberty to define the product of two number-pairs in any convenient manner; but the only definition which does not give rise to results that are merely trivial is that, symbolised' ... and so on. It is hardly fair to a student to expect him to know what the authors consider trivial. ...

Curves were not considered in detail in the first edition; in the second they are considered more at length, but not nearly so carefully as in Mr Watson's Tract referred to above. 'his is not intended to be a reproach; as we know, the careful consideration of what can be meant by a, 'curve' is a much later ,development than Cauchy's theory of functions, and it would seem that a student would be rather puzzled by a complete treatise on analysis situs before he begins to work with complex integration at all. The authors seem to be wise in omitting such a detailed discussion as is given in Mr Watson's Tract. Also, in the fourth chapter of the second edition, on Riemann integration, which the Preface says is entirely due to Mr. Watson, there is a very brief dismissal of the question as to the bounds of a function of n variables which Mr Watson, in his Tract seemed carefully to avoid as if it was something quite different from the question of the bounds of a function of one variable. Perhaps the most important addition to the second edition is the interesting discussion of uniformity and the Heine-Borel theorem. This theorem comes particularly into prominence when we have to prove Cauchy's theorem on complex integration in the manner of Goursat.

**2.2. Review by: Anon.**

*The Mathematical Gazette*

**8**(124) (Jul., 1916), 306-307.

In the first edition of this work, which was written by Prof Whittaker alone and published in 1902, it was stated that the first half was devoted to "those methods and processes of higher mathematical analysis which seem to be of greatest importance at the present time," while in the second half "the methods of the earlier part are applied in order to furnish the theory of the principal functions of analysis." This description holds for this second edition. The original preface is not given in the new edition, and the book has grown by nearly 200 pages. Mr. Watson has added entirely alone the fourth, eleventh, and thirteenth chapters, and has worked with Prof Whittaker at many of the others. Thus the tenth, nineteenth, and twenty-first chapters are presumably added by the labour of both Prof Whittaker and Mr. Watson, and the eighth chapter is a new one, but there was a chapter in the first edition on " Asymptotic Expansions." Besides this, the first few chapters are greatly altered from the first edition. On the whole, the alterations are for the better, as the important questions lying at the bottom of Goursat's proof of Cauchy's theorem, about which the first edition was notoriously defective, are here treated with precision. With regard to the rest of the changes in the first few chapters, which are chiefly prompted by advances in modern logic, it cannot be said that they are by any means satisfactory either to a learner or to a teacher. But this does not affect the more important-at least in the authors' eyes-part of the book, and can be, moreover, easily rectified by a study of such a book as Mr Hardy's

*Course*.

**2.3. Review by: Anon.**

*The Monist*

**26**(4) (1916), 639-640.

The first edition (by Professor Whittaker alone) of this work was published in 1902, and in the preparation of the second edition Professor Whittaker has been most ably helped by Mr Watson. To Mr Watson the new chapters on Riemann Integration, Integral Equations, and the Riemann Zeta-Function are practically wholly due. Part II ("The Transcendental Functions") is, as we should expect, most admirably done; but, since the subject-matter is exclusively technical, the philosopher and logician will turn with more interest to those chapters in Part I ("The Processes of Analysis") in which more fundamental subjects are discussed. It is a most pleasing fact that the treatment of irrational numbers, the theory of convergence, and the proof of the theorem of Cauchy and Goursat on complex integration by the help of the "modified Heine-Borel theorem," are so well done in this new edition. The theorem attributed to Bolzano was not really proved by Bolzano. Bolzano used, in 1817 and not in 1851 as stated, the same process which afterwards, in the hands of Weierstrass, led to an exact proof. The exact proof of the condition mentioned on page 14 is also due to Weierstrass and not to Cauchy. The book is a thoroughly good one, and will be of great value in English and American universities.

**2.4. Review by: Anon.**

*Science Progress (1916-1919)*

**11**(41) (1916), 160-161.

It is scarcely necessary to give a formal review merely on the occasion of a second edition of so admirable and well known a book as this one; but, as we happen to know, many have been long awaiting the second edition, and our readers may therefore be glad to be furnished with some idea as to the changes and additions made in it. In the first place, for the second edition, Prof Whittaker's name is associated with that of Mr G N Watson, who, Prof Whittaker tells us in his Preface, is responsible for the new chapters on Riemann Integration, on Integral Equations, and on the Riemann-Zeta Function. The second edition adopts the Peano decimal system of paragraphing, much to the advantage of the work. The two Parts of the original work still stand; but the original Chapter 4 on Uniform Convergents is suppressed and partly replaced by the new Chapters 3 and 4 on Continuous Functions and Uniform Convergents and on the Theory of Riemann Integration. The old chapters 3, 5, 6, 7, and 8 still stand with changed numbers. But three more new chapters have been added to the First Part, namely on Fourier's Series, Linear Differential Equations and Integral Equations.

**3. A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (3rd edition 1920), by E T Whittaker and G N Watson.**

**3.1. Contents.**

Part I. The Processes of Analysis.

Chapter I: Complex numbers;

Chapter II: The theory of convergence; I

Chapter II: Continuous functions and uniform convergence;

Chapter IV: The theory of Riemann integration;

Chapter V: The fundamental properties of analytic functions; Taylor's, Laurent's, and Liouville's theorems;

Chapter VI: The theory of residues; application to the evaluation of definite integrals;

Chapter VII: The expansion of functions in infinite series;

Chapter VIII: Asymptotic expansions and summable series;

Chapter IX: Fourier series and trigonometrical series;

Chapter X: Linear differential equations;

Chapter XI: Integral equations.

Part II. The Transcendental Functions.

Chapter XII: The gamma function;

Chapter XIII: The zeta function of Riemann;

Chapter XIV: The hypergeometric function;

Chapter XV: Legendre functions;

Chapter XVI: The confluent hypergeometric function;

Chapter XVII: Bessel functions;

Chapter XVIII: The equations of mathematical physics;

Chapter XIX: Mathieu functions;

Chapter XX: Elliptic functions. General theorems and the Weierstrassian functions;

Chapter XXI: The theta functions;

Chapter XXII: The Jacobian elliptic functions;

Chapter XXIII: Ellipsoidal harmonics and Lame's equation;

Appendix;

List of authors quoted;

General index.

**3.2. Review by: Anon.**

*The American Mathematical Monthly*

**28**(4) (1921), 176.

The first edition of this work was by Whittaker alone in 1902. The second edition in collaboration with Watson appeared in 1915. "Advantage has been taken of the preparation of the third edition ... to add a chapter on Ellipsoidal Harmonics and Lame's Equation, and to rearrange the chapter on Trigonometrical Series so that the parts which are used in applied mathematics come at the beginning of the chapter. A number of minor errors have been corrected and we have endeavoured to make the references more complete" (Preface).

**3.3. Review by: Eric Harold Neville.**

*The Mathematical Gazette*

**10**(152) (1921), 283.

Encyclopaedic in scope, this treatise is one of the of the notable achievements of Cambridge scholarship in our time, and the title is as apt now as it was in 1902. But in this edition only two chapters call for comment, for elsewhere the work is essentially a reprint of the edition of 1915, and the authors have not attempted to meet criticisms: for example, not only is the clumsy theory of real numbers in the first chapter repeated, but it is still described as Russell's.

The chapter on trigonometrical series has been rearranged and revised. By the revision, in which in particular the proof of Fejér's fundamental theorem has been simplified, we can all benefit. The rearrangement is for the sake, we are told, of students interested primarily in applications, but it is possible to doubt whether physicists and applied mathematicians will turn even now to these pages to make the acquaintance of Fourier series for the fascination that made the first edition of

*Modern Analysis*a delight to schoolboys and a purifier of the mechanically-minded has evaporated.

The other substantial change is the addition of an irritating and inspiring chapter on ellipsoidal harmonics and Lame's equation. The elementary formulae relating to confocal coordinates might surely have been quoted, since the variables are restricted quite unnecessarily to be real; nothing either novel or elegant is said on this head. In both manner and matter the analysis leading to the construction of ellipsoidal harmonics is childish in the extreme ... The second half of this chapter shows the authors at their best bringing to bear on the solution of Lame's equation a great variety of methods, and making a skilful combination of results due to a number of independent workers. And the student to whom earlier chapters have conveyed the impression that nothing in analysis remains to be done will learn at last of one equation in which a parameter has not yet been generalised.

**3.4. Review by: Dorothy Maud Wrinch.**

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

**15**(60) (1921), 658.

There are few changes in the new edition of this book. It still holds the field as the repository of information about the transcendental functions for the applied mathematicians, and it has come to be considered as a standard reference book of the methods of the theory of functions. It is at once plain that any book which achieves both these admirable objects only manages it through a perpetual struggle in the mind of the authors. There are evidences of this struggle throughout the book. It is not impossible that a further edition might most profitably introduce sweeping changes in the direction of redrafting the initial chapters (and in particular the very first, which is not up to the level of the rest of the book) in the interests of the mathematician, and developing in a somewhat gentler and less aristocratic manner the vagaries of the transcendental functions and integral equations, Fourier Series, and other mathematical phenomena which are so often a dark puzzle (and a painful one) to physicists who, for no fault of their own, sooner or later have to deal with them. The rearrangement of the chapter on Fourier Series, which is a step in this direction, is therefore to be welcomed.

An interesting chapter on Lamé's functions, which is entirely new and has not appeared in the previous editions, renders accessible much of the recent work on ellipsoidal harmonies which sooner or later must be of considerable importance in mathematical physics.

**4. A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (4th edition 1927), by E T Whittaker and G N Watson.**

**4.1. Review by: Anon.**

*The Mathematical Gazette*

**14**(196) (1928), 245.

The fourth edition of this standard treatise does not differ much from the third. All the misprints with which the writer was previously acquainted appear to have been corrected; and the list of references is more complete. There are no new chapters, or discussions of new types of special functions. The general excellence of the work, which stands quite alone, is so well recognised everywhere that a new edition, in these circumstances, leaves little for a reviewer to say.

But we still have certain regrets, from the point of view of the applied mathematician, regarding the fundamental introduction of the special functions as contour integrals rather than from differential equations. This, however, is no hardship to the mathematical analyst, for to him it gives a comprehensive outlook on the whole class of functions as special cases of a very general theory of analysis, which it is the fundamental aim of the book to expound. This is the reason why, as the authors say, the Jacobian elliptic functions, for example, are theoretically best visualised as quotients of Theta-functions - but not, for instance, by a student of higher dynamics, or allied subjects, who wishes to express his solutions in a canonical shape in terms of the Jacobian functions. As a teacher of mathematics, the writer has been driven to conclude that such students frequently cannot bear to " learn " the Theta-function theory as an approach to the functions $sn, cn, dn$, which are quite simple generalisations of trigonometric functions, and subject to tabulation.

Perhaps, in view of the aim of the work, this concession should not be made, but it was done in the original Whittaker's "Modern Analysis" preceding these four editions, and the change is a trouble to many students.

**4.2. Review by: Allen Stenger.**

*Mathematical Association of America*(16 January 2009).

https://www.maa.org/press/maa-reviews/a-course-of-modern-analysis

Alf van der Poorten accurately summarised this book in his Notes on Fermat's Last Theorem by writing, "Notwithstanding its title, 'Whittaker and Watson' is neither 'modern' nor 'analysis,' as we now understand it. But it is the bible of the classical special functions."

The last edition of this bible was in 1927, and it is still in print from Cambridge. Although it covers a lot of ground, its focus is primarily on the differential equations of mathematical physics and the special functions that are their solutions. This is not a mathematical physics book - it quotes the differential equations and states where they arise, but does not derive them or show how to solve them. The book is notable for having a large number of very challenging exercises (here called "Examples," in the traditional English style).

The book is divided in two parts. Part I, "The Processes of Analysis," starts at the beginning and presents all the background needed for Part II, "The Transcendental Functions." Part I covers a large swath of classical analysis in only 250 pages, including the Riemann integral, infinite series, analytic functions, quite a lot about Fourier series, and some about differential and integral equations. It is not comprehensive on any of these subjects, but it covers the basics thoroughly and some of the more advanced parts. The treatment is straightforward and not too different from what is in most contemporary introductory texts on these subjects.

Part II, at about 350 pages, gives an almost-comprehensive account of the special functions of mathematical physics, and of a number of other special functions that are used more generally, for example, the gamma function and the elliptic functions. There is a relatively skimpy chapter on the Riemann zeta function. Most or all of these functions have individually been the subjects of whole books, but Whittaker and Watson give you coverage that is almost as complete, and often easier to use, in only 30 or 40 pages. Part II, used as a reference rather than as a text, is the real strength of this book.

**5. A Treatise on the Theory of Bessel Functions (1922), by G N Watson.**

**5.1. From the Preface.**

This book has been designed with two objects in view. The first is the development of applications of the fundamental processes of the theory of functions of complex variables. For this purpose Bessel functions are admirably adapted; while they offer at the same time a rather wider scope for the application of the parts of the theory of functions of a real variable than is provided by trigonometrical functions in the theory of Fourier series.

The second object is the compilation of a collection of results which would be of value to the increasing number of Mathematicians and Physicists who encounter Bessel functions in the course of their researches. The existence of such a collection seems to be demanded by the greater abstruseness of properties of Bessel functions (especially of functions of large order) which have been required in recent years in various problems of Mathematical Physics.

While my endeavour has been to give an account of the theory of Bessel functions which a Pure Mathematician would regard as fairly complete, I have consequently also endeavoured to include all formulae, whether general or special, which, although without theoretical interest, are likely to be required in practical applications, and such results are given, so far as possible, in a form appropriate for these purposes. The breadth of these aims, combined with the necessity of keeping the size of the book within bounds, has made it necessary to be as concise as is compatible with intelligibility.

**5.2. Review by: Milton Brockett Porter.**

*The American Mathematical Monthly*

**30**(6) (1923), 326-327.

Bessel functions are perhaps from the point of view of applied mathematics the most important transcendants after the exponential group and their inverses.

It is this importance that justifies this great treatise on their theoretical properties and while the applied mathematician might have wished for illustrations of their use in physical problems, such additions would have vastly increased the size of the book and proved disappointing to those largely interested in the theoretic aspects of these functions. That Professor Watson has examined a large amount of material is shown by the index of titles of more than seven hundred papers and an author index of more than three hundred names ...

Professor Watson deserves the gratitude of his colleagues for the able and careful manner in which he has bought together this large amount of material as well as for the well-balanced his own personal investigations. It is unnecessary to say that the book work of the Cambridge Press is beyond all praise and is absolutely unrivalled elsewhere, fine paper, beautiful printing and wide margins only marred by a somewhat flimsy binding. The book should certainly find a place, despite the high price, in every good mathematical library, but the cost will doubtless prevent its purchase by many persons desiring to own a copy of it.

**5.3. Review by: F P W.**

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

**18**(70) (1923), 304-306.

Prof Watson has produced a monumental treatise which is not likely to be superseded for years to come. It is large, expensive, and full of good stuff. It is very unlikely that any known formula relating to Bessel functions is not given somewhere in the book, but it is perhaps doubtful whether the mathematical physicist who has need of it will be able to find it in finite time. His attempt, however, will be profitable and instructive; his eye will be caught by some interesting theorem wherever he may happen to open the book, and he will read on, fascinated, forgetting perhaps his original quest, until either the subtlety of the logic or the complication of the formulae becomes too much for him. And he will not only get information about Bessel functions, he will also find brief accounts of various branches of theory which are not given in the ordinary textbooks, but which are necessary for special applications. ...

One may, perhaps, be allowed to express a regret that Prof Watson has not made a thorough examination of the symbolic methods of Heaviside instead of contenting himself with the remark: "It is difficult to decide how valuable such researches are to be considered when modern standards of rigour are adopted."

**5.4. Review by: Anon.**

*The Mathematical Gazette*

**18**(231) (1934), 349-350.

Prof Watson's treatise was a predestined classic on the day on which it appeared, though no one would have ventured to predict that it would be readable as well as authoritative. The absence of a review in the

*Gazette*at the time is explained by a paragraph on the last page of vol. xi: "The Editor will be grateful if the member to whom he sent, on its publication, a copy ... will kindly inform him of its whereabouts." But why Mr Greenstreet failed to make the usual memoranda in this case remained a mystery, and no delinquent ever claimed the promised gratitude.

A bibliography of 38 pages, and 87 pages of numerical tables, many of them calculated for this volume, give some indication of the scale on which Prof Watson planned his work and of the immense industry which he devoted to it. But ambition and industry, though not always combined with accuracy, are comparatively common. It is in the 650 pages of mathematical exposition that Prof Watson's rarer qualities are revealed. An encyclopaedic wealth of know- ledge of the subject and of whole realms of theory which it illustrates, infallibility in logic, lucidity in argument, all compel admiration. On the vexed question of the choice of canonical functions of the second kind, Prof Watson writes fairly and persuasively, and it is to be hoped that for the sake of uniformity his decisions will be accepted in the future even by writers, if any there are, who are not convinced by what he says.

While the author's outlook is that of the pure mathematician, he has "endeavoured to include all formulae, whether general or special, which, although without theoretical interest, are likely to be required in practical applications."

**5.5. Review by: R D Carmichael.**

*Bull. Amer. Math. Soc.*

**30**(1924), 362-364.

The purpose of this book is twofold: to develop certain applications of the fundamental processes of the theory of functions of complex variables for which Bessel functions are admirably adapted; and secondly, to compile a collection of results which shall be of value to the in- creasing number of mathematicians and physicists who encounter Bessel functions in the course of their researches. The author believes that the existence of such a collection is demanded by the greater abstruseness of properties of Bessel functions (especially of functions of large order) which have been required in recent years in various problems of mathematical physics.

In his exposition the author has endeavoured to accomplish two specific results: to give an account of the theory of Bessel functions which a pure mathematician would regard as fairly complete; and to include all formulas, whether general or special, which, although without theoretical interest, are likely to be required in practical applications. An attempt is made to give the latter results, as far as possible, in a form appropriate for use in the applications. These exalted aims the author seems to have achieved with a remarkable success. The great amount of material thus to be included and the necessity for keeping the size of the book within bounds have made necessary the employment of a concise and compact exposition: but this has been attained without undue sacrifice of clarity. Nearly all parts of the book are as easily read as one has a right to expect in the case of material of this sort. The general theory on which the special results in this volume are based is to be found in the Course of Modern Analysis by Whittaker and Watson. It is greatly to the reader's convenience to have a single volume for reference for the basic theory on which the exposition is founded.

...

The author's work is well done throughout. The reviewer did not detect any omissions or serious blemishes of any kind. In some respects the exposition is probably more disjointed than is necessary. Some parts would profit by being brought into closer connection with related general theories relative to differential equations; but the author is right in avoiding the use of any large part of the general theory of differential equations. It appears to the reviewer that the exposition of the theory of the zeros of Bessel functions would profit by a closer connection with the general oscillation theory of the solutions of linear homogeneous differential equations of the second order, especially since that theory has taken such simple form in the hands of Bôcher. In a similar manner the general notions involved in the Birkhoff theory of expansions in terms of orthogonal and biorthogonal functions satisfying linear differential equations would serve to give greater unity to the four long chapters on expansions in terms of Bessel functions. But, even so, there is something to be said in favour of the direct treatment employed by the author.

Many of the results recorded appear to be of the nature of special instances of general propositions yet to be discovered. In the case of these a disjointed exposition is inevitable until the general theory has been brought into existence. Perhaps the book will serve a third important purpose in addition to the two which the author had in mind in preparing it. It is suited to suggest further researches in two ways: it will raise interesting questions concerning Bessel functions and the differential equation which they satisfy; it will also suggest general theories concerning linear differential equations, theories special instances of which are afforded by the Bessel equation. In this respect the expansion theory seems to hold out the greatest promise. We may well close the review by emphasising the importance in this respect of the four excellent chapters devoted to expansions of functions in series of Bessel functions.

**6. A Treatise on the Theory of Bessel Functions (2nd edition, 1944), by G N Watson.**

**6.1. Review by: Harry Bateman.**

*Science, New Series*

**101**(2614) (1945), 117-118.

This excellent book was written at a time when the author was much interested in the propagation of electromagnetic wave over the surface of the earth, and consequently one of the important features of the book is that it contains material of interest to the radio engineer. Such a man is interested particularly in the asymptotic expansions of the Bessel functions, in definite integrals involving Bessel functions and in tables of Bessel functions. The subject of asymptotic expansions is treated with the thoroughness characteristic of a master in this field. It may be recalled that in 1912 Watson published in the Rendiconti di Palermo a memoir crowned by the Danish Royal Academy of Science in which among other things he gave expressions for the functions J_n(x), J_{-n}(x), Y_n(x) and K_n(x) as series of inverse factorials.

**6.2. Review by: Harold Thayer Davis.**

*National Mathematics Magazin*e

**19**(3) (1944), 153-154.

One cannot but marvel at the abounding richness and variety of mathematics when he reflects that the contents of this great treatise flow from the functions defined by the linear differential equation:

$x^{2}y'' + xy' + (x^{2} - n^{2})y = 0$.

This equation is called the Bessel equation and its solutions, Bessel functions, after F W Bessel (1784-1846), who, although he was not the first writer to use the functions named after him, gave their principal properties and constructed the first tables of $J_{0}(x)$ and $J_{1}(x)$ in a lengthy memoir on planetary perturbations published in 1824. As a matter of historical fact, the function of zero order, $J_{0}(x)$, was used by Daniel Bernoulli in 1732, and the functions of first kind, among functions $J_{n}(x)$, by L Euler in 1764 and by J L Lagrange in 1769.

The functions thus defined were soon found to have applications in many fields of science. In their varied uses, they rival the circular functions. They are found in physics, astronomy, chemistry, geodesy, geology, and even in a science as remote from their origins as economics. The literature of the subject during the last century has become very voluminous. For example, Watson records in his bibliography 797 books and memoirs published prior to 1922. Some 431 tables of the Bessel functions, and of functions intimately related to them, have been computed, many of them since 1922.

The treatise of Watson has stood for nearly a quarter of a century as the definitive reference work on Bessel functions. The author himself contributed more than a dozen original investigations to the subject. Moreover, from a long series of studies on the general character of asymptotic series, he was peculiarly equipped for an analysis of the intricate problems associated with the asymptotic expansions of the functions and of their zeros. One of the first accounts ever published in English on the powerful method of steepest descents (also called the saddle-point method) is found in his treatise.

It is unfortunate, however, that this second edition of the classical work on Bessel functions is not a real revision of the first. As the author says: "To in- corporate in this work the discoveries of the last twenty years would necessitate the rewriting of at least Chapters XII-XIX; my interest in Bessel functions, however, has waned since 1922, and I am consequently not prepared to under- take such a task to the detriment of my other activities." The revision consists merely of the correction of a few minor errors and the emendation of a few assertions, upon which research since 1922 has thrown additional light.

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

*The Mathematical Gazette*

**29**(283) (1945), 37-38.

This long-awaited volume is effectively a photographic reprint of the original edition of 1922, with misprints and small errors corrected. The author tells us that his interest in the subject has waned since 1922, and that therefore he has not been prepared to undertake the task of rewriting half the book to incorporate the results of many workers obtained during the last twenty years. It cannot be any reproach against the author if we remark on the great disappointment which this decision will cause; he is no doubt right in his refusal to sacrifice to this task something of his many other activities. But on the other hand we are equally right in regretting that we shall not now have his precise and masterly marshalling of results and formulae accumulated since 1922. This is perhaps particularly to be regretted in connection with the long chapter on infinite integrals involving Bessel functions, so closely related to the Laplace operational calculus, and productive of so many curious and interesting formulae. For instance, some paragraphs presenting in an orderly fashion the relations between Bessel functions and the polynomials of Laguerre and Sonine, would have been most helpful; and there must be many other developments of which the same remark could be made.

Discussion of the book an author ought to have written is pointless; but a re-issue of a classic is hard to review. "Watson" is a classic; and as of a result in analytical conics we say "It's bound to be in Salmon" or of a dynamical problem "It's bound to be in Routh", so of any result about Bessel functions we say, with perhaps even greater confidence, "It must be in Watson". Mathematicians, physicists, electrical engineers have all relied on finding therein the results they wanted, and have rarely been disappointed; its value may be judged, crudely but effectively, by the very high prices commanded by second-hand copies of the first edition. Clearly, for once at least, the stock phrase of meeting a long-felt want is thoroughly appropriate to this new edition; and in spite of war-time difficulties, the Cambridge Press has been able to turn out a volume in the best style of their "big blue books", even the paper, though not quite of peace-time standard, being sufficiently opaque to prevent the print showing up on both sides. It may be noted that the price is 10s. less than that of the original edition. It is a pleasure to know that one of the most important and authoritative English mathematical treatises of the inter-war period is once more available.

**6.4. Review by: Marion Cameron Gray.**

*Quarterly of Applied Mathematics*

**2**(4) (1945), 356.

For those already familiar with Watson's "Bessel Functions" a new edition needs no recommendation, and indeed the fact that a second edition has appeared in the midst of wartime difficulties is in itself sufficient evidence of the book's importance. Only a few minor corrections have been made at this time, but it is still cause for rejoicing that "Watson" is once more readily available. In the preface to the new edition the author regrets that his interest in Bessel functions has waned since the book was first published, and rightly remarks that to have included all the new material which has appeared in the last twenty years would have meant rewriting most of the book. It is unfortunate that he did not find it possible to bring at least the bibliography up-to-date so that for recent work we must still look elsewhere.

Nonetheless, the book remains an invaluable compendium of information for any mathematician whose work ever touches on Bessel functions. While the presentation is mostly designed for the pure mathematician the applications are not completely neglected, and the applied mathematician will find it particularly useful if he regards it as a treatise on the theory of functions of a complex variable, with applications to Bessel functions.

If there are any applied mathematicians whose work has not so far brought them into contact with "Watson" a brief list of some of the topics treated may prove illuminating. Such a list, by no means complete, would include: solutions of Riccati's equation, expansion of functions in series of Bessel functions (including Fourier-Bessel, Dini, Kapteyn and Schlömilch series), addition theorems, methods of evaluating definite and infinite integrals, asymptotic expansions for Bessel functions of large argument and of large order (using the principle of stationary phase and the method of steepest descents), determination of the zeros of Bessel functions, and last but by no means least, tables of various Bessel and related functions.

Both author and publisher are to be congratulated on the successful reissue of this classic treatise. May it long remain in print.

Last Updated September 2020