Customer Reviews

Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals

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This is a very engaging introduction to the wonderful world
of integrals from many perspectives. It is very elegantly
written. It has a chuckfull of ideas for projects for
undergraduates, with or without computer algebra.
I strongly recommend it to everyone from smart
sophmores to full professors, in all areas of mathematics.

This book is a gem. It is full of beautiful integral and series formulas for classical mathematical constants, with lucid explanations of how to derive them. There are many examples of using the symbolic language Mathematica to generate conjectures. Connections with number theory are emphasized and the reader will find a glimpse into the magic world of Ramanujan.
The main goal of the book is to evaluate integrals, including those of Euler and Laplace. Surprisingly, other topics covered are: Fibonacci and Bernoulli numbers, generating and hypergeometric functions, solutions of 3rd and 4th degree equations, irrationality, the constants of Apery, Catalan and Euler, recursions, the Prime Number Theorem and the Riemann zeta function. The last chapter is an introduction to the revolutionary WZ method of summation.
The book is written at the level of a college junior and would make an excellent textbook for a course. Lovers of mathematics will read it for pleasure. (Full disclosure: the book mentions results of the reviewer.)

This book is a miscellany of all kinds of different definite integrals, considered from many aspects. It often considers two different evaluations of an integral, leading to an identity between the two answers. This can provide an explicit formula for an infinite sum that would have been impossible to discover directly. Quite a lot of the material comes from the pages of the American Mathematical Monthly, a repository of thousands of clever proofs.

Among the charms of this book are the provision of many different proofs for well known formulas, for example for the Wallis formula for pi, for zeta(2), and for the normal integral, int exp(-x^2) = sqrt(pi). I thought the normal integral proofs were especially fascinating. The book starts with the familiar proof by squaring the integral to get a double integral then switching to polar coordinates, then proceeds through some quite startling methods, including several that tie it to the Wallis formula. There's even a "number theory" method based on the number of representations of a number as a sum of two squares!

The earlier portions of the book emphasize guessing and heuristics, especially by using Mathematica or a similar program to calculate particular values that can be examined for a pattern. This aspect diminishes as the book progresses; the latter sections are more of a straight "theorem-proof" exposition without much motivation or discovery. The exposition is clear throughout, but I agree with Polya that we should "teach guessing" and I was disappointed that this book couldn't sustain its good start.

This is a good book for browsing. Don't try to read it straight through--just leaf through the pages and stop when you see something interesting.

Everyone who has touched calculus and analysis during the very first university years knows how much time must be `lost' in the justification of quite clear facts (almost obvious to our intuition). At the same time, one becomes familiar with several tricky counterexamples to those clear facts if a condition (looking rather inessential!) is missed. Arriving then at the subject of definite and indefinite integration, one usually starts to think of calculus as a boring but unavoidable part of mathematics, and it is only on a higher level of university study when one may taste the real beauty of analysis by seeing so many wonderful applications of those boring statements.

It is astonishing to see in the book "Irresistible integrals" that even the first steps in calculus may be full of an irresistible grace and that many contemporary beautiful results may become available so early. It is the task of the authors to introduce the reader to the subject of analysis by means of methods of definite integration and their applications to mathematical functions. The methods involved are not only classical (purely analytic) but also modern, like symbolic evaluation in computer algebra systems. An important novelty of the book is demonstrating the role of experiment in mathematics. This is really something that may help the beginner to understand the basic ideas of proofs and evaluations before doing the proofs rigorously and the evaluations formally. The book contains a lot of exercises, examples and computer algebra programs. Complex analysis is completely avoided to make the book as elementary as possible. Among the applications, the reader will find many results on special mathematical functions and mathematical constants.

Do not think that the book can be used only by those who are starting the study of higher mathematics or who have missed a reasonable introduction to calculus. "Irresistible Integrals" is also nice reading for experts in analysis, number theory, combinatorics and algorithmic theory. The authors have themselves participated in developing new evaluation and transformation techniques for definite integrals, which the reader may now get at first hand. I have no doubts in forecasting that this book will not be buried in the reader's bookshelf.

Victor Moll and the late George Boros have really excelled themselves in producing Irresistible Integrals: it is really replete with gems. The book contains a multitude of fascinating formulae and helps to explain the intimate connections between, inter alia, the Riemann zeta function, the Gamma function and Euler's constant. Everything is set out well and explained with great clarity.
This book could be read and appreciated by a very diverse audience: in the UK, a good A level maths student would have his eyes opened wide by the wonderful theorems placed before him and an aspiring PhD will become so enthused that one day he will win the $1million prize for proving the Riemann Hypothesis!
More seriously though, I highly recommend this book and look forward to seeing Volume II in print. Read it and enjoy.

Using integration techniques is an area of undergraduate mathematics that has been shunted along the neglected road of late. With the advent of other areas of mathematics and the increased use of symbolic mathematics packages, it is possible to teach concepts that could not be offered a few years ago. However, a great deal can be learned from the underlying principles of integration, and this book can be used to teach them.
The authors state that their target level is the advanced undergraduate student and they achieve that goal. Most of the concepts can be understood by anyone who has successfully completed the three-course sequence in basic calculus. Some of the topics are integrals involving binomial coefficients, partial fractions, trigonometric functions, quartic integrals, Euler's constant, the gamma and beta functions and the Riemann zeta function.
The authors use Mathematica commands to perform many of the operations and those commands are easy to understand. Furthermore, the use of Mathematica is as a supplement, it would not be necessary to understand a single Mathematica command in order to be able to follow all of the demonstrations.
I rarely teach math classes any more, and those that I do teach are at the lower levels. However, if I was given the opportunity to teach a special topics class in mathematics, I would seriously consider offering one in the fundamental techniques of integration and use this as the text.

Two of my students and I have been using this book this year for an independent study project. We have found the material quite fascinating and we have learned a lot. The material is well-organized and, as we approach the last chapter, it becomes clear what the authors' overarching theme was.
The exercises range from trivial to challenging. We skipped most of the projects, but it seems like they are quite challenging as well.
However, the book is full of typographical errors, many of them significant. This has been an unending source of frustration for us. Nothing is more aggravating than to spend an hour trying to prove some statement that turns out to be false as written, only to find out that there is a missing factor or minus sign. Chapter 11 alone has at least 9 such errors among its 18 pages.
So I recommend this book if you are interested in learning more about integrals, but be prepared to check every statement and do not assume the book is correct.

If you want a BOOK on these topics, look for one by Dunham such as
"Euler the master of us all". In particular, Euler was a great writer.