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**Pi and the AGM. A study in analytic number theory and computational complexity.**
*(English)*
Zbl 0611.10001

Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. xv, 414 pp. £48.00 (1987).

The central theme of this book is the efficient calculation of mathematical constants.

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.

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.

This is a delightful book in the classical tradition, full of beautiful formulae, and ably complemented by the excellence of the typography and layout.

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.

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.

This is a delightful book in the classical tradition, full of beautiful formulae, and ably complemented by the excellence of the typography and layout.

Reviewer: H.J.Godwin

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11Y60 | Evaluation of number-theoretic constants |

11F03 | Modular and automorphic functions |

68Q25 | Analysis of algorithms and problem complexity |

33E05 | Elliptic functions and integrals |

65B99 | Acceleration of convergence in numerical analysis |

65D20 | Computation of special functions and constants, construction of tables |

11J81 | Transcendence (general theory) |

33B10 | Exponential and trigonometric 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### Digital Library of Mathematical Functions:

§19.35(i) Mathematical ‣ §19.35 Other Applications ‣ Applications ‣ Chapter 19 Elliptic Integrals§19.5 Maclaurin and Related Expansions ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM) ‣ §19.8 Quadratic Transformations ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals

§23.20(iv) Modular and Quintic Equations ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions