NA TURE 



169 



THURSDAY, JUNE 22, 1893. 



THE THEORY OF FUNCTIONS. 

 Theory of Functions of a Complex Variable. By Dr. A. 



R. Forsyth. (Cambridge University Press, 1893.) 

 Al/H'^T is the theory of functions about ? This ques- 

 ^^ tion may be heard now and again from a mathe- 

 matical student ; and if, by way of a pattial reply, it be 

 said that the elements of the theory of functions forms 

 the basis on which the whole of that part of pure mathe- 

 matics which deals with continuously varying quantity 

 rests, the answer would not be too wide nor would it 

 always imply too much. 



It cannot be denied that the teaching of pure mathe- 

 matics in this country has followed curiously restricted 

 lines. While in geometry and the theory of forms the 

 student has for many years past had the advantage of 

 excellent English text-books, the general theory of 

 functions has been entirely unrepresented till the appear- 

 ance of the treatise whose title stands at the head of this 

 notice. Of treatises on special classes of functions, if we 

 omit those written purely with a view to applications. Cay- 

 ley's " Elliptic Functions," published in 1876, is the sole 

 representative ; while till last year there was no work on 

 the theory of numbers. The theory of groups, and its 

 applications to the theory of equations, is still unrepre- 

 sented in native English mathematical literature, though 

 here we have the translations of Prof. Klein's " Vorlesun- 

 gen iiber das Icosaeder," and Herr Netto's " Substitution- 

 entheorie," published, the one in 1888, and the other last 

 year. At Cambridge, and probably to a great extent in 

 other centres, the teaching and the course of study of 

 individual students have tended on the whole to follow 

 the lines of the available English text-books, and where 

 these have been incomplete or entirely wanting there has, 

 till very recent years, been no sufficient introduction to the 

 corresponding subjects. 



Why a subject of such fundamental importance for the 

 advancement of pure mathematics as the theory of 

 functions should have happened to fall into this latter 

 class, it is not easy to tell. It may be said to have been 

 first put on a secure footing by Cauchy's great memoir on 

 integrals taken between imaginary limits, which was pub- 

 lished in 1825. Many advances were made by a number 

 of eminent mathematicians in the following years, and 

 the study of the subject received a great impetus from 

 the new and very fascinating method of presenting it 

 which Riemann gave in his famous memoirs on the theory 

 of functions of a complex variable (1851), and on the 

 theory of the Abelian functions (1857). 



Weierstrass and his pupils, again, developed their 

 theory from a standpoint which is essentially distinct 

 from that of either Cauchy or Riemann. The growth 

 of the subject during the last thirty years has been 

 remarkable, and it is probably safe to say that the 

 foreign literature of the subject is now more extensive 

 than that of any other branch of pure mathematics. 

 The number of text-books that have been published 

 directly on the subject is wonderful in itself, and more so 

 when it is remembered that almost every foreign treatise 

 on the Differential and Integral Calculus contains some 

 introduction to the theory of functions. 

 NO. 1234. VOL. 48] 



If there is any justice in the preceding remarks, the 

 want of a treatise on this subject has too long caused a 

 serious gap in our mathematical literature ; and it may 

 be at once said that Dr. Forsyth's book supplies that 

 want so completely that it is not likely to be felt again 

 for a long time to come. 



Among the large number of foreign treatises above 

 referred to are several which, in their own line, it would 

 be difficult to improve upon ; but they all, or nearly all, 

 deal with the subject from a single point of view, being 

 indeed written with that intention. Dr. Forsyth, on the 

 other hand, has aimed at giving a complete introduction 

 to the theory ; and it may safely be said that, with his 

 book as a guide, the task of the student who wishes to 

 enable himself to follow its various recent developments 

 will have lost half its difficulty. 



The bringing of the various parts of the subject, and 

 the different points of view from which they may be 

 approached, into their proper connection with each other 

 has here been done in the most masterly way ; while 

 though Dr. Forsyth expressly disclaims in the preface 

 to have dealt at length with anything but the general 

 theory, he has carried the developments of the subject 

 in the direction of doubly-periodic and allied functions, 

 Abelian integrals, and automorphic functions to a point 

 from which the student can have no difficulty in passing on 

 to the study of any recent work done in these branches. 



It is impossible in the limits of a short article to 

 give any complete account of a book extending to 

 over 700 pages, but some attempt may be made 

 to describe the order of treatment. The first four 

 chapters are devoted to the simpler properties of uni- 

 form functions, their expansion in power-series and 

 their integration. Chapters v., vi. and vii. deal with uni- 

 form transcendal functions, giving the principal results 

 of the investigations of Weierstrass and Mittag-Leffler. 

 In this connection the very remarkable result due to 

 Weierstrass is given, which is expressed by him in the 

 following words : — " Dass der Begriff einer monogenen 

 Function einer complexen Veriinderlichen mit dem 

 Begriff einer durch arithmetische Grossenoperationen 

 ausdriickbaren Abhangigkeit sich nicht voUstandig 

 deckt." The writer of a recent criticism in this journal 

 would probably say that this statement deals only with 

 the morbid pathology of mathematics ; but the pure 

 mathematician at all events should surely know, as far 

 as possible, what is implied in the word function. 



Non-uniform functions are introduced in chapter viii. 

 They are regarded, to begin with, as arising from the 

 various continuations of a power-series, the most general 

 point of view that can be taken ; Riemann's method of 

 dealing with algebraic functions and their integrals not 

 being introduced till considerably later. The following 

 chapter deals with the integrals of non-uniform functions ; 

 and from the particular examples given arise some of 

 the simplest singly- and doubly-periodic functions, whose 

 properties, when uniform, are discussed in Chaps, x., xi., 

 and xii. This part of the subject aptly ends with a 

 demonstration, due to the author, of the theorem that if 

 /{"), f{v), and/(a + w) are connected by an algebraical 

 equation with constant coefficients, /(») must be either 

 an algebraic, a simply-periodic, or a doubly-periodic 

 function of u. The proof of this important theorem by 



I 



