Differential Forms in Mathematical Physics. by C. Von WestenholzReview by: Robert HermannSIAM Review, Vol. 22, No. 1 (Jan., 1980), pp. 106-107Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2029892 .

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106 BOOK REVIEWS

In Chapter 10 the author treats what is to the reviewer the most intriguing aspect of Bessel polynomials, namely, the location of their zeros in the complex plane.

By now the following facts are known: i) the zeros of yn(Z, a) are simple; ii) no zero of y, is a zero of yn+i; iii) for a ? 2 the zeros lie strictly in the left-half plane. The most recent bound for their moduli is due to Saff and Varga: for Re a > 1 - n,

the zeros lie inside the cardioid r = (1 -cos O)/(n + a - 1) and outside the circle r= [n (n + a - 1)-'.

For the case a = 2 asymptotic formulas for the zeros can be obtained from Olver's work on uniform asymptotic expansions of Bessel functions: the zeros lie asymptotically on a half-eye in the left-half plane whose parametric equation is known. Everyone suspects that much the same is true for all a but for general a the differential equation apparently cannot be put in a form suitable for the application of Olver's turning point theory.

Sometime ago Luke conjectured that the root of largest magnitude of Yn (Z; a) satisfies

A -2 1.32548n + (a- 1)-

This conjecture has not yet been resolved. The asymptotic properties of the polynomials themselves are easy to establish

(Chapter 13):

Yn (z; a) ( ) 2a3/2 e 1z{1 + O(n -')}, n - oo

uniformly on compact z-sets excluding the origin. What are lacking are uniform asymptotic estimates valid near z - 0.

Chapters 11 and 12 treat the algebraic properties of Bessel polynomials. It is known that all Bessel polynomials of degree ?400 are irreducible. Whether they are all irreducible is an open question.

Chapter 14 is devoted to applications, including the previously mentioned ones. There is a very satisfying discussion of the inversion of Laplace transforms.

The number of references, 185, is amazing for a book of only 174 pages. The reviewer's own work on the zeros of Bessel polynomials was buried in a paper whose title held no clue to this endeavor-yet the author found it out. The book deserves to be called an encyclopedia of Bessel polynomials.

My dissatisfactions with the book are minor; and most can be traced to what I call my notational solipsism: the notation of all mathematicians other than myself is lousy. For instance, I prefer to see more use made of hypergeometric notation, which can reveal otherwise hidden patterns. The author gives a notational index but fails to indicate on which page the symbol was first defined: an inconvenience.

This very welcome volume is surprisingly attractive and readable for a quick-copy book.

JET WIMP

Drexel University

Differential Forms in Mathematical Physics. By C. VON WESTENHOLZ. North- Holland, Amsterdam and New York, 1977. xv+487 pp. $65.25. Dfl. 150.00.

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BOOK REVIEWS 107

Differential geometry is both one of the oldest and newest, the most pure and the most applied, branches of mathematics. Throughout most of its history it has had close contact with physics; the exception was the post-Relativity period (say, 1935-1970) when the pure tendencies predominated and it became a poor relation of topology and analysis. However, in recent years there has been an exciting return to the more traditional sources of inspiration, andthere are now extremely active areas of research concerned with the interaction between differential geometry, physics, control theory, continuum and fluid mechanics, nonlinear field theories, chemistry and biology.

The creation of a new, broadly-based applied differential geometry has been hindered by the lack of an introductory treatise which is accessible to a mathematically trained scientist or engineer, or to a mathematician who wants to see physical concepts expressed in modern terms. This gap has been partly filled by this book; it is by far the best I have ever seen in this field, and shows excellent taste and judgement.

Its coverage is actually a good deal broader than the title would indicate. Here are the chapter heas1ings: Topological preliminaries, Differential calculus on R', Differen- tiable manifolds, Differential calculus on manifolds, Lie groups, Fiber bundles, Basic concepts of differential forms, Frobenius theory, Integration of differential forms, De Rham cohomology, Connections on fiber bundles, Hamiltonian mechanics and geometry, General theory of relativity.

Reading this book brings up the old and profound questions of the meaning of the distinction between pure and applied mathematics. One hundred years ago, both were done in the spirit of this book. The separate existence of SIAM and AMS testifies to the paradise we have lost. I want to suggest to the readers of this journal that applied mathematics has been prevented from taking its rightful place in the contemporary scientific and technological world by the parochialsim of our separate "pure" and "applied" bureaucracy. The tragedy is that there is no home in our academic or scientific structure for work in the tradition exemplified here.

I hope that the publisher will either lower the price or make a paperback edition so that it can be put to work in the education of a new generation of pure-applied scientists!

ROBERT HERMANN Harvard University

Oligopoly and the Theory of Games. By JAMES W. FRIEDMAN, North-Holland, Amsterdam. 1977. xii+311 pp. This is a careful, useful, worthwhile contribution to the development of oligopoly

theory which is, nevertheless, unfortunately limited. Little that is basically new is presented. But much that has not been previously organized, brought together and presented coherently is set down in this book. This will serve as a useful reference for advanced students and researchers for many years.

The book is divided into two parts: Part I, Traditional Oligopoly Models and Their Extensions and Part II, Game Theory. The second part is naturally divided into four chapters on noncooperative equilibria and two chapters on cooperative game theory applications to oligopoly.

Of the twelve chapters, the first five present a nice, terse historical summary of much work in oligopoly theory. In particular, Cournot is covered excellently and the general conditions for the existence of a Cournot equilibrium are interwoven with a careful historical discussion. The treatment of price-strategy oligopoly of Bertrand, Edgeworth, Hotelling, Chamberlin, Shubik and others is formally correct but not

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