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Pi and the Arithmetic Geometric Mean: Study in Analytic Number Theory and Computational Complexity (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts) Hardcover – 4 Feb 1987

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More About the Author

Jonathan Michael Borwein, FRSC, FAAAS, 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 Rodes 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 ( and a 2010 book on Convex Functions---and over 350 refereed publications.

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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

--This text refers to the Paperback edition.

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Abstract. The focus of this chapter is the arithmetic-geometric mean (AGM) iteration of Gauss, Lagrange, and Legendre and its relationship to elliptic integrals. Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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4 of 4 people found the following review helpful
Valuable, unique 10 Nov. 2010
By Jim Curry - Published on
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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