- Review from the American Journal of Physics
- Review from the Bulletin of the American Mathematical Society
- Review from the Mathematical Reviews

The following review appeared in the

**Handbook of Integration**. Daniel Zwillinger. 367 pp. Jones and
Bartlett Publishers, Boston, Massachusetts, 1992. Price: $49.95
ISBN0-86720-293-9. (Reviewed by David M. Cook.)

As conveyed by its title, this book is neither a text book to be studied from cover to cover nor a table of integrals to be consulted in the hope that someone has already evaluated the current nasty integral. Instead, its objective is to help those who already know some of what they are doing to identify (or remember?) potentially productive approaches and apply them. Rather than list answers, the handbook catalogs methods, including both illustrative examples and voluminous references to the literature. The first chapter illustrates the wealth of contexts in which integrals arise, thereby motivating the remainder of the book. Sections on exact analytic evaluation (two chapter), approximate analytic evaluation (mainly an assortment of methods for finding and assessing asymptotic expansions but including a section on establishing bounds, all presented in one chapter), and numerical evaluation (two chapter) complete the book.. Despite its brief treatment of each individual topic, the book does admit that particular methods don't always work, that numerical approaches must be assessed for accuracy, etc. Each topicisadequately discussed to alert the reader to limitations and sometimes subtle pitfalls. Section 14, titles ``Caveats'' and presenting only three straightforward--though wrong--evaluations, should be required reading for all whose work involved integration.While neither slighting time-honored, traditional approaches nor overlooking the many integrals that define special functions, this book does pay more than a little attention to the role that computers can play in addressing integrals. Indeed the

firstmethod suggested for theanalyticevaluation of integrals is to use a symbol manipulating computer program, and many pages are devoted to comparing and criticizing the output produced when a suite of test integrals is submitted to a representative selection of such programs. These programs recognize a wide spectrum of nonelementary integrals, but all have some limitations, and some--here is the author's warning to the unwary--fall victim to the same sorts or carelessness illustrated in Section 14, ``Caveats.'' Similarly, thefirstmethod suggested for thenumericalevaluation of integrals is to use a commercial computer program or subroutine, and several pages are devoted to orienting the reader both to the classifications schemes used for identifying such routines and to the richness of public-domain and commercial software.To illustrate how this book might be used, suppose we are interested in the familiar Gaussian integral

f(x)=\int. If we seek an_{0}^{x}e^{-t2}stanalyticevaluation, we are directed first to try a symbol manipulating computer program. Those available to me (MACSYMA, MAPLE, andMathematics) all return the correct ``evaluation'' in terms of the error function. From one perspective, our task is done, though we don;t yetreallyhave an evaluation; we have merely a recasting of the question from ``What is the value of this integral?'' to ``What is the value of this special function?'' For more information, we look to the index. The entryGaussian integralis not there. The entryerror functionis more productive, yielding four page references, of which the first is uninformative, the second and fourth present examples but not the function itself, and third--paydirt--identifies the formal definitions of both the error function and the complementary error function and points--with a correct page reference--to detailed discussion in Abramowitz and Stegun. The neighboring entryerror function integralsidentifies a worked-out asymptotic series for thecomplementaryerror function. In hindsight, we would have found the most useful information more quickly had we first recognized that out integral defines a function with one parameter and the scanned that section in the length tabulation of integrals defining special functions (Section 39). Even with a small false start, we are in short order led both to an exact analytic evaluation and an approximate analytic evaluation.If we choose instead a

numericalevaluation, we must find a table of the error function (See Abramowitz and Stegun, to which the search in the last paragraph led us), evaluate the asymptotic series, or seek a numerical evaluation (and worry about the accuracy, whichever route we adopt). The book discusses the essence of a variety of numerical algorithms in some detail, enumerates numerous available items of software (though it mentions the ability of most symbol manipulating programs to evaluate integrals numerically only in a footnote to the section on analytic evaluation), and present ``decision trees'' designed to help us select an appropriate routine for the integrand of interest. Since our test integrand is particularly smooth and well behaved, the simplest methods will work. We need no be concerned about singularities, discontinuities, or oscillating factors in the integrand, though, asxincreases, we might need to contemplate truncation of an integral extending over a nearly infinite interval. True to its title, this book offers suggestions and references to help us deal with these complications. Curiously, I find no mention that symbol manipulating programs often can return numerical values of specific functions at specific arguments.In books of this sort, typographical accuracy is, of course, of particular importance, since most users will not immediately recognize subtle typographic errors in elaborate mathematics formulas. That I discovered only half a dozen typos in several hours of examining this book is, it seems to me, substantial circumstantial evidence that the proofreading has been carefully done. Modern technology does, however, occasionally generate an unusual typographical problem, e.g., the character string

frac12\pii(page 39), revealing both that the book was typeset using LaTeX and that the proofreader failed to notice an omitted backslash in the command $\frac{1}{2\pi i}$.This book contains a rich compendium of information about integral and integration organized in a novel way, and it is well supplied both with references to the literature and with an assortment of guides to help find an appropriate approach, whether the user knows a name of the integral or is searching in the dark. In structure and function it lies somewhere between a standard table of integrals and a standard table of values of special functions. For most of us, I suspect that this book will be about as frequently used as those more conventianal tables; it may well occasionally provide the key pointer that suggests where to look for the information really needed. There is, however, at least some possibility that rapidly advancing computer technology, increasingly reliable and versatile symbol manipulating computer programs, and easily used and readily available routines for numerical integration will soon render

allof these printed reference materials obsolete.

David M. Cook is a Professor of Physics and Philetus E. Sawyer Professor of Science at Lawrence University, Appleton, Wisconsin. His special interests lie in computational physics, nonlinear dynamics, and pedagogical applications of computers.

The following review appeared in

*Handbook of integration*, by Daniel Zwillinger. Jones & Bartlett
Publishers, Boston, MA, 1992, xvi+367 pp., $49.95. ISBN 0-86720-293-9

Reviewed by: Alan Genz, Washington State University

There are now many sources for possible solutions to the diverse collection of integration problems that arise in mathematics. We have the traditional tables of integrals, books on the theory of integration, and books on numerical methods. We also have access to numerical software libraries and symbolic mathematics computing systems. In addition, we also have a large variety of methods that are scattered through books and papers on pure and applied mathematics. This book is an attempt to provide a comprehensive survey of these integration methods. It is a compilation of methods that the author began collecting when he was a graduate student.The book is organized as a collection of eighty-three short sections. These are somewhat loosely arranged under the six chapter headings: Applications, Concepts and definitions, Exact analytical methods, Approximate analytical methods, Numerical methods concepts, and Numerical methods techniques. Each section usually begins with short entries under the categories: applicable to, yields, and idea. These entries are followed by more extensive discussion under the categories: procedure, examples, notes, and references. The references are extensive and up to date. For example, Section 50, entitled ``Stationary Phase'', begins:

Applicable to

Integrals of the formI(\lambda)=\int, where^{b}_{a}g(x)e^{i \lambda f(x)}dxf(x)is a real-valued function.

Yields

An asymptotic approximation when\lambda >>1.

Idea

For\lambda\to\inftythe value ofI(\lambda)is dominated by the contributions at those points wheref(x)is a local minimum.The section continues with a half-page description of the procedure and a half-page example. This is followed by one and one-half pages of notes and ten references. The references range from a 1918 paper by G. N. Watson to a 1991 paper by J. P. McClure and R. Wong and include three books. The section is typical. The mathematics is clear and concise. There is enough information for the reader who wants a short discussion of the topic of interest, and there are good references for someone who wants more detail.

The book is clearly a reference book. It is shorter than what one might expect for a book with a title

Handbook of integration,, but I suspect that most integration problems in applied mathematics would have at least the beginning of a solution outlined in this book. People who have a variety of applied integration problems should find this book to be a valuable reference that is easy to use. I would like to have seen more information about integrals that arise in statistics. There is a brief mention of the one-dimensional normal distribution function but little else. Other readers might also find some of their favorite topics missing, but they are also likely to find plenty of new material and references. There is no other book that provides such a broad and up-to-date survey of integration methods.

The following review is from the

Author: Zwillinger, Daniel

Title: Handbook of integration.

Publication Year: 1992

Publisher: Jones and Bartlett Publishers, Boston, MA, 1992, xvi+367 pp.

Abstract: This is basically a dictionary of mathematical topics relating to integration. The material is organized into six chapters, within each of which the special topics are arranged in alphabetical order. The chapters deal with applications of integration (e.g., extremal problems, geometric applications, probability); concepts and definition of integrals (e.g., exterior calculus, Feynman diagrams, finite part integrals, fractional integration); exact analytical methods (e.g., change of variables, convolution techniques, Frullani integrals, stochastic integration); approximate analytical methods (e.g., asymptotic methods, continued fractions, integral inequalities); principles of numerical methods (e.g., Romberg integration, software libraries, testing of quadrature rules); and specific numerical techniques (e.g., adaptive quadrature, Gaussian quadrature, lattice rules, Monte Carlo methods). Each entry is similarly structured: an indication of its scope, the basic idea, a description of the procedure, examples, notes, and a list of references for further reading. In concept and execution, this work resembles an earlier one by the same author dealing with differential equations [Handbook of differential equations, Academic Press, Boston, MA, 1989; MR 90k:00044; second edition, 1992; MR 92j:00014].