BOOK REVIEWS 441

oscillations in lines; 12. Bessel functions; 13. Small-amplitude vibrations of athread suspended from one end; 14. Small-amplitude radial vibrations of a gas;15. Legendre polynomials; 16. Application of the Fourier method to small-amplitude vibrations of membranes.

Part II: Differential equations of elliptic type. 17. Integral formulae applicableto elliptic equations; 18. Laplace and Poisson equations; 19. Potential theory;20. Logarithmic potential; 21. Spherical functions; 22. Questions on gravimetryand the theory of the shape of the earth; 23. Applications of spherical functionsto problems of mathematical physics; 24. Gravity waves on the surface of aliquid; 25. The Helmholtz equation; 26. The emission and scattering of sound;27. Comments on general equations of elliptic type.

Part III: Equations of parabolic type. 28. Heat-flow equation; general theo-rems; 29. Heat-flow in an infinite rod; 30. Application of the Fourier method toheat-flow problems.

Part IV: Supplementary material. 31. The use of integral operators in solvingproblems in mathematical physics; 32. Application of finite integral transforms;33. Application of integral transformations with infinite limits; 34. Maxwell’sequations; 35. Emission of electromagnetic waves; 36. Directed electromagneticwaves; 37. Electromagnetic horns and resonators; 38. Motion of a viscous fluid;39. Generalized functions.The reviewer found some of the theoretical considerations too sketchy. A

more serious objection" there is almost no mention of the developments in partialdifferential equations during the last twenty years, or of approximate methods.The final chapter on generalized functions contains a number of mistakes. Someminor points: although the reader is told in a footnote when a system of func-tions is called complete, he is left to guess what a closed system is, and why aclosed system is complete. The associated Legendre functions are erroneouslycalled polynomials (p. 335). The translation is reasonably good, even if the.bookcontains a number of "self-conjugate" differential "expressions".

Notwithstanding these shortcomings, the book should make a good text forthe usual junior-senior and beginning graduate course in partial differentialequations for students in physics and engineering.

JACOB KOREVAARUniversity of California at La Jolla

Probability, Random Variables, and Stochastic Processes. By ATHANASIOSPAPOULS. McGraw-Hill Book Co., New York, 1965. xi + 583 pp. $12.75Part I, pp. 3-275, Probability and random variables, constitutes an elementary

introduction to probability for engineers and physicists at an undergraduatelevel. Part II, pp. 279-576, Stochastic processes, is written on a more advancedlevel for applied scientists with some familiarity of the theory of linear systemsand Fourier transforms. Some idea of the topics covered is given by the chaptertitles: Part I, The meaning of probability, The axioms of probability, Repeatedtrials, The concept of a random variable, Functions of one random variable,Two random variables, Functions of two random variables, Sequences of random

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442 BOOK REVIEWS

variables; Part II, General concepts, Correlation and power spectrum of sta-tionary processes, Nonstationary processes (transients in linear systems withstochastic inputs), Harmonic analysis of stochastic processes, Stationary andnonstationary normal processes, Brownian movement and lV[arkoff processes,Poisson process and shot noise. The index covers pages 577 to 583, but there isno collected bibliography. Each chapter is followed by a number of problems,of which the book contains a total of two hundred, and there are as many exam-ples illustrating the results.

In the preface, the author expresses the hope that the book will be judged notfor completeness but for organization and clarity. It is the opinion of the reviewerthat certain topics of importance to engineers and applied scientists could wellhave been included if the book had been differently organized. It seems that thisbook should have been published as two separate volumes in view of the audi-ences to which the parts are addressed. Two separate books, with the more com-plete coverage in each thereby possible, would provide a very necessary anduseful addition to the literature.

In the main. the presentation is clear, logical and reasonably rigorous, but onewonders about references to "the mystery associated with probability" and "inmathematics". It might be useful in referring to works by Kolmogorov and VonMises to also give the reference to translations into English which are available.Incidentally, there is a mispelling in the title of the German reference in thefootnote on p. 260. The spelling "Kinchin" for Khinchin or Khintchine is uniqueto this book. This reviewer found surprisingly few misprints, although the selec-tion of type fonts for some of the symbols could well be improved for clarityand consistence with other works in this field. The author should have used themodern symbols for union and intersection and should have named the sym-metric difference which he defines on p. 3.3. It surely would be useful to the readerto know that the zero-one random variable on p. 89 is also known as an indicatorfunction, and that the generalized Bernoulli trials on p. 74 appear as the multi-nomial distribution. It is difficult to understand why the author has no qualmsabout presenting the so-called generalized function known as the delta or im-pulse function, and yet avoids the Stieltjes integral which plays a very usefulrole in probability and its applications. There were a number of other termsstrange to the reviewer, for example, "staircase function" and the expression"statistics of a random variable y" for the probability function.We quote the author: "The book is written neither for the handbook-oriented

students nor for the sophisticated few who can learn the subject from advancedmathematical texts. It is written for the majority of engineers and physicistswho have sufficient maturity to appreciate and follow a logical presentation, but,because of their limited mathematical background, would find a book such asDoob’s too difficult for a beginning text." In spite of the adverse comments,this reviewer feels that the author, in the main, has consistently followed hisobjectives, and although improvements are still possible and a closer liaison

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with the literature other than engineering and applied science is desirable, thisbook will undoubtedly find a large and receptive body of readers.

SOLOMON KULLBACKThe George Washington University

OTHER BOOKS RECEIVED1. The Gamma Function. By EMIL ARTIN. Holt, Rinehart and Winston, New

York, 1964. vii -t- 39 pp.2. Measure and Integration. By STERLING K. BERBERIAN. Macmillan,. New

York, 1965. xviii - 312 pp. $9.95.3. Advanced Calculus. 2nd. ed. By R. CIGUTO BUCK. McGraw-Hill Book

Company, New York, 1965. ix + 527 pp. $9.75.4. The Dynamic Stability of Elastic Systems. By V. V. BOLOTIN. Holden-Day,

San Francisco, 1964. xii + 451 pp. $12.95.5. Envelopes. Popular Lectures in Mathematics, Vol. 12. By V. C. BOLTYAN-

SKH. Pergamon Press, New York, 1964. ix -t- 76 pp. 82.25.6. Systems and Simulation. By DMTRIS N. CHORAFAS. Academic Press,New York, 1965. xvi -t- 503 pp. $14.50.

7. The Mathematics of Matrices. By PHILIP J. DAVIS. Blaisdell, New York,1965. 368 pp. $7.50. A text for advanced high school students.

8. Tables of Elliptic Integrals of the First, Second and Third Kind. By HENRY E.FETTIS, JAMES C. CASLIN. Aerospace Research Laboratories, Office of Aero-space Research, United States Air Force, Wright-Patterson Air Force Base,Ohio, 1964. iv + 93 pp.

9. The Simplex Method of Linear Programming. By F. A. FICKEN. Holt, Rine-hart and Winston, New York, 1964.

10. Digital Communications with Space Applications. By S. W. GOLOMB, I. D.BAUMERT, M. F. EASTERLING, J. J. STIFFLER, AND A. J. VITERBI. Prentice-Hall, Englewood Cliffs, New Jersey, 1964. xi - 210 pp.

11. Spectral Analysis of Economic Time Series. By C. W. J. GRANGER in associa-tion with M. HATANAKA. Princeton University Press, Princeton, 1964.xviii + 299 pp. $8.50.

12. Groups and Their Graphs. By ISRAEL GROSSMAN AND WILHELM MAGNUS.Random House, New York, 1965. vii + 196 pp. 81.95.

13. Foundations of the Theory of Algebraic Invariants. By G. B. GUREVICH.P. Noordhoff Ltd., Groningen--The Netherlands, 1964. viii -t- 429 pp.$16.25.

14. Similarity Analyses of Boundary Value Problems in Engineering. By AawHUG. HANSEN. Prentice-Hall, Englewood Cliffs, New Jersey, 1965. xiv. + 114pp. 89.00.

15. Nonlinear Oscillations in Physical Systems. By CHINIRO HAYASttI. Mc-Graw-Hill, New York, 1964. xii -t- 392 pp. 814.50.

16. An Introduction to Quantitative Business Analysis. By IRA HonowIwz. Mc-Graw-Hill Book Co., New York, 1965. xii -t- 270 pp. 87.95.

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