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

George Boros (Author), Victor Moll (Author)

Book Description

ISBN-10: 0521796369 | ISBN-13: 978-0521796361 | Publication Date: June 21, 2004

The problem of evaluating integrals is well known to every student who has had a year of calculus. It was an especially important subject in nineteenth century analysis and it has now been revived with the appearance of symbolic languages. The authors use the problem of exact evaluation of definite integrals as a starting point for exploring many areas of mathematics. The questions discussed are as old as calculus itself. In presenting the combination of methods required for the evaluation of most integrals, the authors take the most interesting-rather than the shortest-path to the results. They illuminate connections with many subjects, including analysis, number theory, algebra and combinatorics. This is a guided tour of exciting discovery for undergraduates and their teachers in mathematics, computer science, physics, and engineering.

Editorial Reviews

Review

"This book would be perfect in a basic elective course to hone calculus and proof writing skills and as enrichment reading for talented calculus students." Choice

Book Description

The problem of evaluating integrals is well known to every student who has had a year of calculus. It was an especially important subject in 19th century analysis and it has now been revived with the appearance of symbolic languages. In this book, the authors use the problem of exact evaluation of definite integrals as a starting point for exploring many areas of mathematics, including analysis, number theory, algebra and combinatorics. The questions discussed here are as old as calculus itself. This will be a guided tour of exciting discovery for undergraduates and their teachers in mathematics, computer science, physics, and engineering.

Product Details

• Paperback: 320 pages
• Publisher: Cambridge University Press (June 21, 2004)
• Language: English
• ISBN-10: 0521796369
• ISBN-13: 978-0521796361
• Product Dimensions: 9 x 6.1 x 0.8 inches
• Shipping Weight: 14.4 ounces (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (8 customer reviews)
• Amazon Best Sellers Rank: #489,454 in Books (See Top 100 in Books)

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17 of 18 people found the following review helpful:

5.0 out of 5 stars Irresistible Mathematics, July 27, 2004

By 

D Zeilberger (Princeton, NJ) - See all my reviews
(REAL NAME)   

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This review is from: Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals (Paperback)

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.

13 of 13 people found the following review helpful:

5.0 out of 5 stars A Gem, September 23, 2004

By 

Jonathan Sondow (New York, NY) - See all my reviews

This review is from: Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals (Paperback)

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.)

12 of 12 people found the following review helpful:

4.0 out of 5 stars Good for browsing, June 8, 2005

By 

Allen Stenger (Alamogordo, NM USA) - See all my reviews
(REAL NAME)   

This review is from: Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals (Paperback)

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.