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Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity Paperback – July 13, 1998

ISBN-13: 978-0471315155 ISBN-10: 047131515X

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Product Details

  • Paperback: 432 pages
  • Publisher: Wiley-Interscience (July 13, 1998)
  • Language: English
  • ISBN-10: 047131515X
  • ISBN-13: 978-0471315155
  • Product Dimensions: 6.1 x 0.8 x 9.2 inches
  • Shipping Weight: 1.6 pounds (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Best Sellers Rank: #3,290,159 in Books (See Top 100 in Books)

Editorial Reviews

From the Publisher

Presents new research revealing the interplay between classical analysis and modern computation and complexity theory. Two intimately interwoven threads run though the text: the arithmetic-geometric mean (AGM) iteration of Gauss, Lagrange, and Legendre and the calculation of pi[l.c. Greek letter]. These two threads are carried in three directions. The first leads to 19th century analysis, in particular, the transformation theory of elliptic integrals, which necessitates a brief discussion of such topics as elliptic integrals and functions, theta functions, and modular functions. The second takes the reader into the domain of analytic complexity--Just how intrinsically difficult is it to calculate algebraic functions, elementary functions and constants, and the familiar functions of mathematical physics? The answers are surprising, for the familiar methods are often far from optimal. The third direction leads through applications and ancillary material--particularly the rich interconnections between the function theory and the number theory. Included are Rogers-Ramanujan identities, algebraic series for pi[l.c. Greek letter], results on sums of two and four squares, the transcendence of pi[l.c. Greek letter] and e[ital.], and a discussion of Madelung's constant, lattice sums, and elliptic invariants. Exercises.

From the Back Cover

Critical Acclaim for Pi and the AGM:

"Fortunately we have the Borwein's beautiful book . . . explores in the first five chapters the glorious world so dear to Ramanujan . . . would be a marvelous text book for a graduate course."—Bulletin of the American Mathematical Society

"What am I to say about this quilt of a book? One is reminded of Debussy who, on being asked by his harmony teacher to explain what rules he was following as he improvised at the piano, replied, "Mon plaisir." The authors are cultured mathematicians. They have selected what has amused and intrigued them in the hope that it will do the same for us. Frankly, I cannot think of a more provocative and generous recipe for writing a book . . . (it) is cleanly, even beautifully written, and attractively printed and composed. The book is unique. I cannot think of any other book in print which contains more than a smidgen of the material these authors have included.—SIAM Review

"If this subject begins to sound more interesting than it did in the last newspaper article on 130 million digits of Pi, I have partly succeeded. To succeed completely I will have gotten you interested enough to read the delightful and important book by the Borweins."—American Mathematical Monthly

"The authors are to be commended for their careful presentation of much of the content of Ramanujan's famous paper, 'Modular Equations and Approximations to Pi'. This material has not heretofore appeared in book form. However, more importantly, Ramanujan provided no proofs for many of the claims that he made, and so the authors provided many of the missing details . . . The Borweins, indeed have helped us find the right roads."—Mathematics of Computation

More About the Author

Jonathan Michael Borwein, FRSC, FAAAS,FBAS, FAA is currently Laureate Professor in the School of Mathematical and Physical Sciences at the University of Newcastle (NSW). He directs the University's Priority Research Centre in Computer Assisted Research Mathematics and its Applications (CARMA).

A Rhodes Scholar, his research interests span pure (analysis), applied (optimization), computational (numerical and computational analysis) mathematics, and high performance computing. He has authored over a dozen books---most recently four on Experimental Mathematics (www.experimentalmath.info), a 2010 prize winning book on Convex Functions, a 2012 book on Lattice Sums and a 2014 book on Continued Fractions, and two on Modern Mathematical Computation---and over 400 refereed publications.

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Most Helpful Customer Reviews

4 of 4 people found the following review helpful By Jim Curry on November 10, 2010
Format: Paperback
This book fills an important and very neglected place in the mathematics curriculum. We suppose we understand the usual, foundational transcendental functions, like the logarithm, exponential, trig functions, and several others. Still, most of us don't really know practical approaches to computing the values of these functions efficiently. We may punch a calculator button or even look at an old tables book, but we ourselves don't always have good ways to compute the values from scratch. If you consult most numerical analysis books, they will give approaches to other computations and even approaches to computing higher transcendental functions, but these foundational calculations are neglected. So, this book is practically the only careful and complete record in book form of the ways these computations ought to be done.

These topics are often bypassed in the curriculum as offered in most schools---even decent schools. So, there are a lot of students (like me) who will want to consult this book as a way to fill in glaring holes in the existing education. It is, of course, entirely possible to use this as a successful text for its own special topics course. That is seldom done, but it would be reasonable. The writing is clear. The scholarship is excellent. There is no reason to complain about this book. The only people who won't just LOVE this book are those who don't care too much for writing that has lots of technical detail. If you're a "big picture" person and don't enjoy knowing the details that make an area craftsmanlike, then you might not enjoy this book as much as I do. I do love it entirely, and feel grateful it is available. Finally, I understand this isn't a great review. I just couldn't stand to see such a fine book with no reviews on Amazon. I invite you to write a more particular review and replace mine.
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