BOOK REVIEWS 759

prised at this choice of emphasis. Indeed, we should be grateful for so thoroughand (as much as possible) elementary discussion of this difficult area.

SEYMOUR V. PAINTERUniversity of Wisconsin

Constructive Real Analysis. By ALLEN A. GOLDSTEIN. Hrper & Row, New York,1967. xii d- 178 pp. $9.25.This is one of a new class of books which is beginning to appear, bringing to

bear on concrete problems of applied mathematics the results obtained in the lsthalf century or so of abstract analysis. In the present cse, the ideas exploitedare from functional analysis (including the differential calculus of nonlinearoperators) and convexity; the dedication of the book is, appropriately, to L. V.Kantorovich and V. Klee.The book is designed to make the material accessible to the mixed under-

graduate-graduate level and to theoretically inclined physical scientists s wellas mathematicians.The author has chosen as his central theme the location of roots of functions

and functional equations and the associated question of finding extremls invariational problems. The theory is exposed first for n-dimensional spaces,studying functional iteration, fixed-point theorems, gradient methods, steepestdescent, Newton’s method and convex mathematical programming. In the lastthird of the book infinite-dimensional spaces are introduced, and generalizationsof these methods are studied and applied. Problems scattered throughout thebook continue the theory and point toward the areas of application.At the end of the book, the student applied mathematician would be properly

motivated to study functional analysis, not merely as an interesting way ofmingling algebra, analysis and geometry, but also as a useful and almost essentialtool in such fields as numerical analysis and optimization theory. The physicalscientist would have a collection of methods useful in many problems, togetherwith a motivation to learn more than is contained in classical courses in "Methodsof Mathematical Physics."The execution is precise but terse; the ground is covered in a no-nonsense,

to-the-point fashion and leaves the instructor much room for amplification, tyingthe course into the folk-knowledge of the student. The physical appearance ofthe volume is very good.

ARTHUR WOUKNorthwestern University

Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equa-tions. By A. W. BABISTER. The Macmillan Co., New York, 1967. xi + 414pp. $14.95.This book will probably be ignored by pure mathematicians. It will appeal

only to those applied mathematicians who are willing to share the author’sidle fixe..The subject is as quaint and improbable as the title of the book itself,

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760 BOOK REVIEWS

and the author pursues it armed only with the most ordinary ot weapons and arelentless preoccupation with detail.The first two chapters are devoted to the elementary theory of linear differen-

tial equations. This reviewer wonders why. The material can be found in anyof the number of standard books on the subject, and much of it is irrelevant towhat follows.

In Chapter 3, the author discusses Struve and Lommel functions. Almostall of this material can be found in either the Bateman manuscript volumes,Higher Transcendental Functions and Tables of Integral Transforms (McGraw-Hill), or in Yudell L. Luke’s book, Integrals of Bessel Functions (McGraw-Hill),which, strangely enough, is not included in the uthor’s bibliography.The fourth chapter treats nonhomogeneous confluent hypergeometric func-

tions. The development in this chapter, t least in he last half, is fairly typicalof the author’s pproch. First, he takes a differential equation which occursfrequently in applied mathematics (here, the confluent hypergeometric equa-tion) and then appends to the right-hand side an algebraic (nd sometimes anexponential) term, so that we have, in this cse, the equation

(1) L(D)y zy" + (c z)y’ ay z-.Two linearly independent solutions of (1) are denoted by O(a, c; z) and 0(a,

c; z), and for them the author determines series representations, recurrenceformulas, integral relations and particular values.

Later chapters contain a similar account of the differential equations satisfiedby parabolic cylinder functions, functions of the paraboloid of revolution, theGaussian hypergeometric function, Legendre’s functions, Lam6 and Mathieufunctions, and finally by the generalized hypergeometric function. Thus theauthor, engages in a sort of expositional overkill: Whittaker’s equation in itsvarious forms and special cases, for example, is treated at least a half-dozen times.This practice can be justified when formulas hold for special cases which cannotbe generalized, but here it seems the formulas derived and the approach usedare virtually independent of the differential equation. Indeed, one of the author’smost wearisome idiosyncrasies is to work from the special case to the moregeneral, which only serves to emphasize .the caprice with which the material wasselected. For instance, immediately after he discusses (1), the author turns to

(2) L(D) ez-.Why? One feels the author might answer, "Why not?"The book closes with two chapters on. partial differential equations. These are

a potpourri of disjecta membra, much of it known (e.g., the separation of La-place’s equation, in various coordinate systems). Again, there seems to be no uni-lying purpose behind the presentation.Each chapter contains an extensive set of problems but, alas, the book itself

is mostly mere formulae-spinning, so what conceivable facility are the exercisesmeant to develop? In fact, the purpose of the book is not at all clear. The re-viewer doubts that the subject is of sufficient importance to justify the writing

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BOOK VWS 761

of a textbook, and most of the new results in the book could have been workedout by any good student with a first course in complex analysis behind him.

JET WIMPThe Mathematical InstituteEdinburgh, Scotland

The Logic of Decision. By RICHARD C. JEFFREY. McGraw-Hill Book Co., NewYork, 1965. xiv -- 201 pp. $7.95.This review discusses another philosophical treatise on. subjective probabilities,

subjective utilities and Bayesian decision making.The principal difference between the theory advanced by the author and other

approaches to the problem is as follows: Given any set of propositions along withtheir subjective probabilities and desirabilities, the author requires that the agentbe able to assign subjective probabilities and subjective desirabilities to anyproposition that can be formed from these alternatives using the logical con-

nectives, conjunction, disjunction, and denial, in any combination, and thisassignment must be consistent with a set of axioms given in the fifth chapter.This means that the decision maker (or agent) must be capable of assigningthese numbers to a bewildering variety of alternatives. Thus, to this reviewer,it is not clear that this possesses any advantages over more conventional theories,in which the agent is to assign utilities to probability mixtures of alternatives,and from these assignments the subjective probabilities are determined. How-ever, the author’s theory does lead to some interesting consequences and a fewof these will now be described.

In more conventional theories, such as Ramsey’s, the preference ranking ofalternatives determines the subjective probabilities uniquely, and the desir-abilities are determined up to a change of location and scale. In the author’stheory, two probability and desirability assignments, denoted by (PROB,DES) and (prob, des), respectively, are compatible with the same preferenceordering of propositions,

PROB X prob X(c des X -- d)

DES X adesX -t-bcdes X + d’

where ad bc > O, c des X + d > 0 for every proposition X, and c des T + d1, where T is the necessary proposition (i.e., universally true).Another novel feature is that if the desirability assignments are unbounded

either above or below, but not unbounded in both directions, then, in general,there is a bounded set of desirability assignments which retains the same prefer-enee ordering for all propositions. The author exploits this in claiming a "resolu-tion of the St. Petersburg paradox." However, this reviewer found his interpre-tation of this resolution somewhat uneonvineing.

In addition, while the preferenee orderings do not determine the probabilities

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