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Zbl 0611.10001
Borwein, Jonathan M.; Borwein, Peter B.
Pi and the AGM. A study in analytic number theory and computational complexity. (English)
[B] Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. New York etc.: John Wiley \& Sons. xv, 414 pp. \sterling 48.00 (1987).

The central theme of this book is the efficient calculation of mathematical constants. \par A brief sketch of the contents is as follows. In chapters 1 and 2 the arithmetic-geometric mean is defined and its connection with elliptic integrals and theta functions is shown. In chapter 3 Jacobi's triple product is introduced and applied to theta functions and in other ways. Chapter 4 gives higher order transformations and modular functions, and chapter 5 uses the previous material to obtain algebraic approximations to $\pi$. In chapter 6 the complexity of computational methods is discussed, and in chapter 7 the complexity of algorithms applied to particular functions is dealt with. Chapter 8 introduces general means, chapter 9 gives various applications of theta functions, and chapter 10 gives methods for accelerating the convergence of classical methods of calculation of various functions, especially exp and log. Chapter 11 gives a history of the calculation of $\pi$, and a discussion of transcendence and irrationality. An extensive bibliography follows. Many results in the text are given as exercises for the reader to prove. \par Aside from the main course of the book there are interesting digressions into, for example, results on representation as sums of squares, series that enumerate partitions, and lattice sums that arise from chemistry. \par This is a delightful book in the classical tradition, full of beautiful formulae, and ably complemented by the excellence of the typography and layout.
[H.J.Godwin]

MSC 2000:: *11-02 Research monographs (number theory)
11Y60 Evaluation of constants
11F03 Modular and automorphic functions
68Q25 Analysis of algorithms and problem complexity
33E05 Elliptic functions and integrals
65B99 Acceleration of convergence
65D20 Computation of special functions
11J81 Transcendence (general theory)
33B10 Elementary functions

Keywords: calculation of mathematical constants; arithmetic-geometric mean; elliptic integrals; theta functions; modular functions; approximations to $\pi $; complexity of algorithms; convergence; bibliography; sums of squares; partitions; lattice sums

Cited in: Zbl 1229.11165 Zbl 1178.26003 Zbl 1203.11088 Zbl 1007.39015 Zbl 1004.26020 Zbl 1110.11300 Zbl 0986.11045 Zbl 0903.11001 Zbl 0893.11001 Zbl 0948.65017 Zbl 0821.33011 Zbl 0733.58027 Zbl 0707.65027 Zbl 0672.10017 Zbl 0652.10019 Zbl 0699.10044

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