Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity / Edition 1

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New York, NY 1987 Hardcover American First Very Good+ No Jacket as Issued Book Second printing; minor wear; otherwise a solid, clean copy with no marking or underlining; ... collectible condition. Read more Show Less

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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.
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Table of Contents

Ch. 1 Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration
Ch. 2 Theta Functions and the Arithmetic-Geometric Mean Iteration
Ch. 3 Jacobi's Triple-Product and Some Number Theoretic Applications
Ch. 4 Higher Order Transformations
Ch. 5 Modular Equations and Algebraic Approximations to [pi]
Ch. 6 The Complexity of Algebraic Functions
Ch. 7 Algorithms for the Elementary Functions
Ch. 8 General Means and Iterations
Ch. 9 Some Additional Applications
Ch. 10 Other Approaches to the Elementary Functions
Ch. 11 Pi
Symbol List
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