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Jonathan M. Borwein
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Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (Wiley–Interscience and Canadian Mathematics Series of Monographs and Texts) Hardcover – 4 Feb. 1987

by Jonathan M. Borwein (Author), Peter B. Borwein (Author)
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This book presents new research revealing the interplay between classical analysis and modern computation and complexity theory. Two intimately interwoven threads run through the text: the arithmetic-geometric mean (AGM) iteration of Gauss, Lagrange, and Legendre and the calculation of pi. 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, results on sums of two and four squares, the transcendence of pi and e, and a discussion of Madelung's constant, lattice sums, and elliptic invariants. Exercises are also included.
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  1. ISBN-10
    0471831387
  2. ISBN-13
    978-0471831389
  3. Publisher
    Wiley–Blackwell
  4. Publication date
    4 Feb. 1987
  5. Language
    English
  6. Dimensions
    16.9 x 2.8 x 24.2 cm
  7. Print length
    432 pages
  8. See all details
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Product details

  • Publisher ‏ : ‎ Wiley–Blackwell (4 Feb. 1987)
  • Language ‏ : ‎ English
  • Hardcover ‏ : ‎ 432 pages
  • ISBN-10 ‏ : ‎ 0471831387
  • ISBN-13 ‏ : ‎ 978-0471831389
  • Dimensions ‏ : ‎ 16.9 x 2.8 x 24.2 cm

About the author

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Jonathan M. Borwein

Jonathan M. Borwein

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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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