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Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity

ISBN: 978-0-471-31515-5
432 pages
July 1998
Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (047131515X) cover image


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
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Table of Contents

Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration.

Theta Functions and the Arithmetic-Geometric Mean Iteration.

Jacobi's Triple-Product and Some Number Theoretic Applications.

Higher Order Transformations.

Modular Equations and Algebraic Approximations to pi.

The Complexity of Algebraic Functions.

Algorithms for the Elementary Functions.

General Means and Iterations.

Some Additional Applications.

Other Approaches to the Elementary Functions.



Symbol List.

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