REVIEWS 149 



Lecons snr les Fonctions Monogenes uniformes d'une variable complexe. 



By Emile Borel, Professeur de theorie des fonctions a l'Universite de Paris. 

 Redigees par Gaston Julia. [Pp. xii + 165.] (Paris : Gauthier-Villars et 

 Cie., 1917. Price 7 fr. 50c.) 



IN this, the most recent of M. Borel's " Collection de monographies sur la the"orie 

 des fonctions," we get an insight into much of M. Borel's life-work in mathematics. 

 In his Thesis of 1894 he took up the question of the possibility of defining a 

 " continuation " of an analytic function beyond a " natural limit " of essential 

 singularities, the impossibility of which then seemed to have been shown by 

 Poincare. In fact, Poincare had constructed analytical expressions with certain 

 singularities, and had given an argument to prove that the theory of analytic 

 functions could not be extended beyond those kinds of regions defined by 

 Weierstrass. In this book, the Preface contains a short summary of the history of 

 the author's researches ; it was not till 191 2 that he proved that he had succeeded 

 in completely defining more extended regions than those of Weierstrass, in which 

 monogenic functions can be defined by Cauchy's integral which possess the 

 characteristic properties of the analytic functions of Weierstrass. 



It seems, however, not quite justifiable to say (p. viii) that Weierstrass's 

 limitation is, "consequently, purely arbitrary." In fact, Weierstrass's theory 

 concerns power-series and their continuations, and for functions so defined his 

 regions certainly cannot be generalised. It is interesting and important that other 

 functions with the properties mentioned above can be defined, but it is hardly 

 legitimate to speak of the theory of Weierstrass as if Cauchy's theory had gained 

 a victory over it in a new Franco-Prussian conflict. And the author seems to 

 depreciate the merits of Weierstrass, and to attribute to his eminent pupil 

 Prof. Mittag-Leffler opinions (p. vii) which it is incredible that he should hold. 

 Further, it must be remarked that not the least merit of Weierstrass's work on 

 analysis is the thorough treatment of such subjects as irrational numbers, in 

 which, by defining real numbers as classes of rational numbers, he anticipated one 

 of the main points in modern logical theory. 



The first chapter contains the preliminaries of a theory of one-valued functions 

 of a complex variable : Goursat's proof of Cauchy's theorem, a treatment of 

 Weierstrass's regions and the elements of the representation of analytic functions 

 in such regions. This chapter is almost wholly good : the only exception seems 

 to be the often repeated and mistaken views (pp. 3-5) on " finite definability." 

 The second chapter contains the application of Cauchy's integral to the develop- 

 ment in a series of polynomials of a function defined in a region of Weierstrass 

 (Runge's method, and also a modification of some work of Mittag-Leffler). The 

 third chapter contains some remarkable consequences of the development just 

 referred to, and the extension of the theory of analytical continuation. The other 

 two chapters concern the new theory. It is essential that we consider a part of 

 the complex plane containing points everywhere dense in a part of this area ; and, 

 by Chapter IV (cf. pp. 124-6), we may limit our consideration, without loss of 

 generality, to those points with rational co-ordinates. Then the regions (C) aimed 

 at are defined: they are not "perfect" in Cantor's sense, the aggregate of points 

 which is complementary to the part considered of the plane of complex numbers 

 is of measure zero, and there are perfect regions which are analogous to the 

 perfect regions, limited by circular arcs, inside and approximating to a region of 

 Weierstrass (pp. 125, 131). Then monogenic functions are defined in the regions C. 

 In the proof of a theorem analogous to Cauchy's, the hypothesis of the continuity 

 of the derivative is used, although it is "doubtless superfluous" (p. 135). The 



