Ordinary Differential Equations. by Fred Brauer; John NohelReview by: Jagdish ChandraSIAM Review, Vol. 9, No. 2 (Apr., 1967), pp. 264-265Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2027465 .

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264 BOOK REVIEWS

than two, are quoted; it is this reviewer's understanding that questions still exist concerning the validity of these generalizations.

In summary, I do not recommend the book from a theoretical viewpoint. However, for applications, the essence of certain methods for problems in non- linear differential equations may be quickly learned by the presentation and examples given in the text.

HENRY HERMES

Boulder, Colorado

Chebyshev Methods in Numerical Approximation. By MARTIN AVERY SNYDER.

Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1966. x + 114 pp. $7.50. This book provides a survey of methods for obtaining polynomial and rational-

function approximations to continuous functions. Optimum approximation in the uniform (Chebyshev) norm is treated only briefly; the author's emphasis is rather upon expansions in orthogonal polynomials, in Taylor series, and in con- tinued fractions. Techniques for obtaining such expansions are discussed at leingth. Among these one finds, for example, the tau method, Obreschkoff's for- mula, Thiele's continued fraction, and Pade approximation. The problem of modifying an available approximation in order to decrease the computing with- out sacrifice of accuracy is treated in several sections. For example, the economi- zation of power series, Maehly's algorithm for transforming Chebyshev series into rational functions, and the "telescoping" of continued fractions are all described. (In this connection, it would have been helpful to discuss the work (Chiffres 1958-1961) of the late George Hornecker, which deserves to be better known.) A useful feature of the book is the extensive collection of results on Chebyshev polynomials.

E. W. CHENEY

Lund, Sweden

Ordinary Differential Equations. By FRED BRAUER and JOHN NOHEL. W. A. Benjamin, Inc., New York, 1966. vii + 555 pp. This book serves as an excellent link between several elementary textbooks

on ordinary differential equations and an ever mounting number of textbooks and monographs on the advanced theory. In the former class of books there is, naturally, a tendency to stress more on the methods of solutions of special classes of equations. Thus, those who may want further information are obliged to turn to advanced textbooks. An adequate preparation in various disciplines of mathe- matics is essential, however, for a proper grasp of material presented in the latter class of books. This book endeavors to bridge this gap. Throughout the book physical problems are used to motivate the study. At the same time, an attempt is made to explain the need for a unified theory.

The first few chapters present essentially the same material that is usually found in any elementary textbook on the subject. In the later chapters, however, the authors present a lucid (elementary) exposition of stability theory and Lya- punov's second method. Chapter 1 gives several physical examples to motivate a

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BOOK REVIEWS 265

systematic study of differential equations. Various methods of solution are cov- ered in the rest of the elementary chapters (Chapters 2-4). Chapter 5 deals with the systems of linear differential equations with a brief iiitroduction to phase plane analysis for two-dimensional autonomous systems. Chapter 6 treats the classical Laplace transform method of solving initial value problems for linear differential equations. The general existence theory is presented in Chapter 7. Chapter 9 introduces the concept of stability and asymptotic equivalence. Lyapunov's second method and the concept of invariant sets are introduced in Chapter 10. The last section of Chapter 2 and all of Chapter 8 are devoted to the numerical methods of solutions suitable for high-speed computation.

The authors intend to augment Chapter 10 by a treatment of Lyapunov's second method for nonautonomous systems, a discussion of stability under per- sistent disturbances and some applications to control theory and reactor dy- namics. They also propose to add some much needed discussion on periodic solutions with stress on perturbation techniques and Poincare-Bendixson theory of limit-cycles, and a chapter on boundary value problems including self-adjoint- ness, Green's functions, existence of eigenvalues, eigenfunction expansions and Fourier series. In this final form, the book should serve as an excellent self- contained text, especially at an intermediate level. In particular, Chapters 5, 7, 9, 10 and the proposed chapters on periodic solutions and boundary value prob- lems can form a comprehensive introduction to advanced topics in ordinary differential equations. There is, also, adequate material for courses of varying length, and with suitable emphasis the material can be taught at several levels. For instance, the first four chapters (perhaps, together with some more material from Chapters 5 and 8) are suitable for a good introductory course in differential equations. Several examples and exercises are given throughout the book making it even more suitable for classroom adoption.

JAGDISH CHANDRA

Research Laboratory Maggs Research Center Watervliet Arsenal Watervliet, New York

A First Cour-se in Stochastic Processes. By SAMUEL KARLIN. Academic Press, New York, 1966. xi + 502 pp. $11.75. A large number of books on stochastic processes have come out within this

decade. It seems that most of the theory of stochastic processes is covered in these books as far as the mathematical setup is concerned, and there even appear the various applications of specialized processes in a variety of practical fields. However, one might say that there is still a gap between the two major features, namely, theory and application. Those which aim at developing the mathematical theory and structural properties are usually not concerned with discussions of special processes which originate in other fields such as biology, engineering or economics. On the other hand, the books of the other genre deal mainly with illustrating various specific processes and making clear the significant properties

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