REVIEWS 



MATHEMATICS 



Theory of Functions of a Complex Variable. By A. R. Forsyth, Sc.D., 

 LL.D., Math.D., F.R.S., Chief Professor of Mathematics in the Imperial 

 College of Science and Technology, London, and sometime Sadlerian 

 Professor of Pure Mathematics in the University of Cambridge. [Third 

 Edition. Pp. xxiv + 855.] (Cambridge : University Press, 1918. Price 

 30J. net.) 



This book is well known as almost an encyclopaedia of the theory of functions 

 of a complex variable, and the fact that it is now in a third edition bears testimony 

 to the great extent to which it has been used as a textbook. The Preface to the 

 first edition was. written in 1893, and the short description of the contents of 

 the book given there (pp. vi-vii) still remains true : the first part (Chapters I-VII) 

 contains the theory of one-valued functions, beginning with complex integration 

 and going on to the theorems of Weierstrass and Mittag-LefHer ; the second 

 part (Chapters VIII-XIII) contains the theory of many-valued functions, princi- 

 pally after Briot and Bouquet, while a chapter has been devoted to the proof of 

 Weierstrass's results about functions that possess an algebraic addition-theorem ; 

 the third part (Chapters XIV-XVIII) contains a treatment of Riemann's method ; 

 the fourth part (Chapters XIX, XX) treats conformal representation ; and the 

 fifth part (Chapters XXI, XXII) contains an introduction to the theory of 

 automorphic functions. In the second edition, the Preface to which was written 

 in 1900, no substantial alterations were made to the part on the theory of one- 

 valued functions, " but new references are given for the sake of readers who may 

 wish to acquaint themselves with the most recent developments " (p. ix). The 

 other chief alterations in this second edition were in the proof of the existence 

 of various classes of functions upon a Riemann's surface, several additions to 

 Chapter XVIII on algebraic functions and their integrals, and a transference 

 to the second volume of Prof. Forsyth's Theory of Differential Equations of the 

 sections that discussed the properties of certain binomial differential equations 

 of the first order. In this third edition, of which the Preface was written in 1917, 

 some changes have been made, for example, in the establishment of the funda- 

 mental functions in Weierstrass's theory of elliptic functions, a note has been 

 added giving some applications of the theory of conformal representation to 

 some branches of mathematical physics (pp. 639-52), and a note stating the 

 results of the discussion referred to above of certain differential equations. 



It seems unfortunate that in this third edition very little has been done to 

 bring matters up to date, particularly in the first part. Between 1900 and 191 7 

 much has been done towards the putting of Cauchy's theory on a sound basis ; yet 

 here there is no mention of Goursat's remarkable work, no indication is given of 

 modern investigation on " contours," and, indeed, the only marked addition seems 

 to be a reference on p. 49 to the book of Lindelof (1905) on the calculus of 

 residues. One might expect some reference to the work on analytic continuation 



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