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

Borwein, Jonathan M.; Borwein, Peter B.

Softcover

ISBN 10: 047131515X ISBN 13: 9780471315155

Publisher: Wiley-Interscience, 1998

Synopsis

Critical Acclaim for Pi and the AGM:

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

"What am I to say about this quilt of a book? One is reminded ofDebussy who, on being asked by his harmony teacher to explain whatrules he was following as he improvised at the piano, replied, "Monplaisir." The authors are cultured mathematicians. They haveselected what has amused and intrigued them in the hope that itwill do the same for us. Frankly, I cannot think of a moreprovocative and generous recipe for writing a book . . . (it) iscleanly, even beautifully written, and attractively printed andcomposed. The book is unique. I cannot think of any other book inprint which contains more than a smidgen of the material theseauthors have included.--SIAM Review

"If this subject begins to sound more interesting than it did inthe last newspaper article on 130 million digits of Pi, I havepartly succeeded. To succeed completely I will have gotten youinterested enough to read the delightful and important book by theBorweins."--American Mathematical Monthly

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

About this title

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.

About the Author:

Jonathan Michael Borwein was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. Peter B. Borwein is the author of Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, published by Wiley.

About this edition

• Publisher: Wiley-Interscience
• Publication date: 1998
• ISBN 10: 047131515X
• ISBN 13: 9780471315155
• Binding: Paperback
• Number of pages: 432