SIAM REVIEWVol. 5, No. 2, April, 1963

Printed in U.S.A.

BOOK REVIEWS

EDITED BY DONALD GREENSPAN

Publishers are invited to send books for review to Donald Greenspan, Mathematics ResearchCenter, U. S. Army, University of Wisconsin, Madison 6, Wisconsin.

An Introduction to Fourier Analysis. By R. D. STUART. Methuen and Co., Ltd.,London, and John Wiley and Sons, Inc., New York, 1961. 126 pp. $3.00.The usual first course in Fourier Analysis in American engineering schools

gives a fairly rigorous development of the elementary theory of Fourier seriesand integrals and applies them to the solution of classical boundary value prob-lems of mathematical physics. Although Fourier Analysis plays a central rolein electronics and communication theory, these subjects are not mentioned.The present work is entirely in the opposite direction. The treatment of the

essential facts of Fourier Analysis is concise and unrigorous, although the essen-tial ideas of the development are suggested. The range of application to elec-tronics and communications is enormous. In a monograph of such slight dimen-sions most topics can be treated only briefly, but the underlying importance andpower of the methods are shown so clearly that the student will certainly bestimulated to look elsewhere for the details.

D. WATERMANWayne State University

Ordinary Differential Equations. By G. ]IRKHOFF and G. C. ROTA. Ginn and Co.,Boston, 1962. vi + 31.8 pp. $8.50.This book is principally aimed at filling the gap between the numerous ele-

mentary texts on ordinary differential equations and the few rigorous up-to-date presentations of the more advanced parts of the subject. In this aim theauthors have succeeded very well. The exposition, which begins with the usualelementary considerations of first order equations, is soon concerned with topicssuch as first order differential inequalities and the resulting comparison theoremsthat are not commonly found in books of this kind. Yet, throughout, only afai.r knowledge of advanced calculus and of elementary complex function theoryis presupposed, together with some facility for manipulting ectors and matrices.The style is easily readable and unhurried. A good number of examples areworked out as illustrations, and nearly all sections are provided with eercisesof varying degrees of difficulty.

Chapters I through IV deal with first order equations, linear equations ofsecond order with analytic coefficients and their solutions in the form of powerseries, and the general linear equation of nth order with constant coefficients.Included here are sections on transfer functions and the Nyquist diagram whichdo not usually find their way into mathematical texts. The standard existence,uniqueness and continuity theorems for nonlinear systems of first order are

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BOOK REVIEWS 161

given in Chapter V. In Chapter VI autonomous equations are taken up, thephase portrait of two-dimensional systems is discussed in detail, and Lyapunov’smethod is sketched briefly for such systems. Chapters VII and VIII on approxi-mate solutions and efficient numerical integration give the standard numericalquadrature formulas, finite difference methods and the Runge-Kutta methodtogether with error bounds and a discussion of numerical stability. Chapters IXthrough XI are concerned with singular points of second order equations, theclassical Sturm-Liouville theory and eigenfunetion expansions.The present book will no doubt contribute much to the improvement and

modernization of intermediate courses in ordinary differential equations.I-I. A. ANTOSIEWICZUniversity of Southern

California

Introduction to Complex Analysis. By Z. NEtIARI. Allyn and Bacon, Inc., Boston,1961. ix -- 258 pp. 87.50.This well written little book should be favorably received by students and

teachers alike. It makes no pretense at being more than the title indicates, atrue introduction to the subject. One would anticipate its use as a text in a typ-ical three credit course after elementary calculus. The exercises are reasonableand ample. One weakness in the opinion of the reviewer is that analytic con-tinuation is placed in the last chapter titled "Additional Topics," so that the com-pletion of the subject might be missed if the last chapter were omitted. This is noproblem however, as the very least any instructor ever does with any text isrearrange the order and selection of topics to his own taste. The table of con-tents reads: Complex variables, Analytic functions, Complex integration, Appli-cations of Cauchy’s Theorem, Conformal Mapping, Physical Applications, andAdditional Topics. The chapter on physical applications of course is used topresent the basic ideas of two dimensional potential theory including the Green’sfunction, and further examples of conformal mapping.

W. I. THICKSTUNU. S. Naval Ordnance Laboratory

The Calculus of Variations. By N. I. AKHIEZER (translated from the Russian byA. I-I. ]RINK). Blaisdell Publishing Co., New York, 1962. vii + 248 pp.For a long time the choice of English language textbooks for an introductory

course in this subject has been very limited. The present book is a welcome addi-tion. The chapter headings are as follows: (1) Equations of the Calculus ofVariations. (2) Theory of Fields. (3) Generalizations of the Fundamental Prob-lem. (4) Direct Methods in the Calculus of Variations.The aim of the book is to acquaint the reader with the problems and main

methods of the subject and not to serve as a treatise. The first two chapterspresent in detail the simplest problem in n dimensions, namely, the fixed end-

point problem for the variational integral f(x, Y, Y’) dx where Y Y(z)

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