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THURSDAY, MARCH 22, 191-. 



MA THEM A TICAL ANAL YSIS. 



(i) Functions of a Complex Variable. Being 



part I of vol. ii. By Prof. E. Goursat. 



Translated by Prof. E. R. Hedrick and O. 



Dunkel. Pp. x-f259. (Chicag-o and London: 



Ginn and Co., 1916.) Price us. 6d. 

 {2) Integrales de Lebesgue. Fonctions d' Ensemble. 



Classes de Baire. By C. de la Vallee Poussin. 



Pp. viii+151. (Paris: Gauthier-Villars et Cie, 



1916.) Price 7 fr. 

 I (3) Functions of a Complex TariabZe. By T. M. 

 * MacRobert. Pp. xiv -r 298. (London : Mac- 



millan and Co., Ltd., 1917.) Price 125. net. 



i) T^HIS is a competent translation of the last, 

 ■*- substantially revised, edition of the 

 original. Prof. Goursat needs no introduction to 

 the mathematical public, so we content ourselves 

 with directing attention to some of the features of 

 this admirable course. The definition of "analytic 

 function " coincides with Cauchy's definition of 

 " fonction monog^ne " ; the properties of such 

 functions are developed with great lucidity, and 

 the student is easily led on to such ideas as power- 

 series, circles of convergence, Weierstrass's 

 theory of analytic continuation, conformal repre- 

 sentation, and so on. Riemann's surfaces are 

 alluded to, but not discussed ; the main outline 

 follows Cauchy, and in this we think the author 

 is judicious, because, however useful Riemann's 

 surfaces are by their appeal to iuLuition, they 

 are not easily realised by a l>eginner, and they 

 have to be constructed in ever\- special case by 

 analytical methods. 



Sp>ecially noteworthy is the way in which Prof. 

 Goursat exp)ounds some of the more recent dis- 

 coveries and theorems, such as Weierstrass's 

 factorial formulae, Mittag-Leffler's theorem, 

 functions with natural boundaries (such as 

 Schwarzian functions), and so on. He has not 

 only mastered these subjects, but is able to discuss 

 them in an original and illuminating manner. 

 For example, the brief discussion of Abel's 

 theorem (pp. 244-50) brings out the essential 

 point that if R(x, y) is a rational function of 

 X, y, and {xi, yi) is an intersection of two curves 

 4> = o, ^ = 0, then the sum 2R(xi, yi), taken over 

 all the intersections, is a rational function of the 

 coefficients of <f>, if/. This is the real basis of 

 Abel's theorem; all the rest is detail, which may 

 be troublesome enough in any particular case. 



There is a comparatively long section on 

 elliptic functions ; another on the apolication of 

 them to curves of deficiency i ; Hermite's im- 

 portant theory of "cuts " is explained ; and the last 

 chapter is on functions of several variables. Alto- 

 gether we could not wish for a better handbook 

 for students of function-theory ; it is clear, com- 

 pact, and consistent ; the references are to first- 

 rate authorities, and sufficient to introduce the 

 reader to the original sources. We are very glad 

 that the methods of Cauchv and Hermite are 



given such prominence, because they are of per- 

 manent value, and only require some modifica- 

 tions of minor importance to bring' them up to 

 the current standard of rigour. 



(2) Every now and then an advance in function- 

 theorj' compels us to revise our notion of a de- 

 finite integral. When the nature of Fourier 

 series had been properly understood, Riemann 

 generalised the definition of an integral so as to 

 apply to them, and Dirichlet followed on the same 

 lines. One of the main topics of the present 

 course is a new definition of an integral (the 

 Lebesgue integral) reducing, as the case may be, 

 to the Riemann or the ordinary integral when 

 it exists, but applicable to cases where both the 

 previous definitions are at fault. Although brief, 

 the course is so far self-contained that it ought 

 to be intelligible to a reader who knows little or 

 nothing about the theory of sets or that of trans- 

 finite numbers; the first section deals with 

 measurable sets and their content, and the nature 

 of Lebesgue integrals ; the second with " additive 

 functions of sets"; the third with Baire's classi- 

 fication of functions of sets. The last, so far as 

 we can judge, is an important notion, more or 



! less comparable with Hadamard's classification of 



i whole functions {mutatis mutandis, of course), 



I and adding one more to the family of "well- 



I ordered classes " or sequences. Another point 



{ that cannot fail to strike the reader is the extra- 



j ordinary difference between the properties of open 



and closed sets ; this distinction is not a new 



one, but its importance is becoming more and 



more clear. 



The author is sparing in his use of new 

 j symbols and technical terms; one, however, 

 seems deserv'ing of mention as being likely to be 

 verv convenient. If E is any set and E' its com- 

 plement in the whole field considered, <^, the 

 characteristic function (or characteristic) of E, is 

 defined as being i for any element of E, and o 

 for any element of E'. Much use is made of the 

 theory of lattices, more or less in the manner of 

 Minkowski. It is scarcely necessary to add that 

 the author's treatment is original, even when deal- 

 ing with the discoveries of others, and that he con- 

 tributes much of his own invention. 



Although not quite analogous to the problems 

 of this course, we may give an easy example 

 to show the kind of difficulties with which it 

 deals. Take a real positive variable .v, and de- 

 fine f{x) to be I when x is rational, and 2 when 

 X is irrational ; then f{x) is perfectly definite over 

 any closed interval (a, b), but the integral of 

 f(x) from a to & does not exist, either in the 

 ordinary sense or in that of Riemann, because 

 f{x) has finite oscillation within any interv^al 

 5.V, however small, and the discontinuities are 

 crowded together. We have, however, an upper 

 limit integral 2 (b - a), and a lower limit integral 

 (b-a). 



(3) Mr. MacRobert has written a book that is 

 likely to be ver\- useful; it is not too big, the 

 selection of theorems is judicious, and there is a 

 large number of really instructive examples, both 



XO. 247^, VOL. qqI 



