WHITTAKER and WATSON, A Course of Modern Analysis, Part I. Fundamental operations of analysis, p. 338, M.—L. 1933, price 7 rub. 50 kop.; Part II. Transcendental functions, p. 467, M.—L. 1934, price 6 rub. 75 kop.

Modern Analysis by Whittaker and Watson is a very distinctive book. Judging by its content and by the character of the exposition, this book is intended for a reader familiar with an elementary course in higher mathematics; the English subtitle indicates the two aims of the authors: first, to give a rigorous exposition of those chapters of analysis whose foundation is omitted in an elementary presentation, and of those sections which find no place in an introductory course; second, to acquaint the reader with special functions. In accordance with these two tasks, the book is divided into two parts, published separately in Russian translation.

The first four chapters of the first part contain a clear and scholarly exposition of the foundations of analysis; here are given the theory of irrational and complex numbers, the theory of convergence of numerical sequences based on Cauchy’s criterion, the continuity of functions and the uniform convergence of functional sequences, and the theory of Riemann integration. A rich theory is illustrated by examples; the material is given both in the text itself and in the form of problems proposed to the reader for solution. Chapters 5 and 6 contain a rather complete exposition of the foundations of the theory of analytic functions and of their application to the evaluation of definite integrals.

Chapter 7 is devoted to the expansion of functions in series. Here material is collected which is rarely included in textbooks: along with Euler’s and Lagrange’s series, Darboux’s formula, Schlömilch’s series, etc., are given. The following chapter contains the theory of summation of divergent series; chiefly the theory of asymptotic expansions is presented; it contains a very simple proof of Hardy’s theorem. Next comes the theory of trigonometric series; among the elementary questions set out in this chapter we note Riemann’s theory of summation. Chapter 10 sets forth the elements of the analytic theory of linear differential equations and serves as preparation for the chapters devoted to special functions. The last chapter of Part I contains a brief outline of integral equations.

The first part may be recommended to a physicist or engineer who has taken a brief course in analysis at an institution of higher education—for deepening

mathematical education and for acquaintance with those methods of mathematics which have been developed in recent decades and are proving increasingly important for applications; it provides rich material for graduate students who are not mathematicians but need to supplement their mathematical knowledge. In view of the difference between the structure of the English system of higher education and that of our universities, the book can hardly be used directly as a university textbook.

The second part of the book has a more unified character. As already indicated, it is devoted to transcendental functions. The authors’ own tastes have, of course, made themselves felt here too in the choice of material. Thus, after the \(\Gamma\)-function comes an exposition of the properties of Riemann’s \(\zeta\)-function, which has no relation to equations of mathematical physics, as do most of the “special functions”; the principal role of the \(\zeta\)-function is in investigations in analytic number theory. There follows a coherent exposition of transcendental functions defined by linear differential equations of the second order, with the hypergeometric function taken as the starting point. The chapter on degenerate hypergeometric functions largely contains the studies of Whittaker. It is followed by a chapter devoted to Legendre and Bessel functions. The chapter on Mathieu functions presents Hill’s method for the application of infinite determinants and for finding solutions of differential equations with periodic coefficients. The end of the book contains a detailed theory of elliptic functions and, finally, an outline of Lamé functions.

The unity of plan of the second part of the book entails a certain one-sidedness: presenting the whole theory in the light of the complex variable, the authors pay almost no attention to questions connected with boundary-value problems for the equations of mathematical physics; the theory of integral equations of proper values remains untouched. But along the lines chosen by the authors the material is presented with a completeness wholly possible in a general course; some details are transferred to the exercises or replaced by references to the original papers.

The second part of the book is an indispensable aid for every researcher who, in his work (in physics, mechanics, or engineering), encounters special functions; it acquaints the reader with the theory of the corresponding class of functions and often gives indications of their practical application. In view of the wealth of literature used, the Course of Modern Analysis is also a very valuable reference work on special functions, perhaps the only one of its kind in world literature.

The translation, editing, and technical production of the book are entirely satisfactory.

B. Stepanov