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

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Edition: 1st
Format: Paperback
Pub. Date: 1998-07-13
Publisher(s): Wiley-Interscience
List Price: $223.94

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Summary

Critical Acclaim for Pi and the AGM: "Fortunately we have the Borwein's beautiful book . . . explores in the first five chapters the glorious world so dear to Ramanujan . . . would be a marvelous text book for a graduate course."--Bulletin of the American Mathematical Society "What am I to say about this quilt of a book? One is reminded of Debussy who, on being asked by his harmony teacher to explain what rules he was following as he improvised at the piano, replied, "Mon plaisir." The authors are cultured mathematicians. They have selected what has amused and intrigued them in the hope that it will do the same for us. Frankly, I cannot think of a more provocative and generous recipe for writing a book . . . (it) is cleanly, even beautifully written, and attractively printed and composed. The book is unique. I cannot think of any other book in print which contains more than a smidgen of the material these authors have included.--SIAM Review "If this subject begins to sound more interesting than it did in the last newspaper article on 130 million digits of Pi, I have partly succeeded. To succeed completely I will have gotten you interested enough to read the delightful and important book by the Borweins."--American Mathematical Monthly "The authors are to be commended for their careful presentation of much of the content of Ramanujan's famous paper, 'Modular Equations and Approximations to Pi'. This material has not heretofore appeared in book form. However, more importantly, Ramanujan provided no proofs for many of the claims that he made, and so the authors provided many of the missing details . . . The Borweins, indeed have helped us find the right roads."--Mathematics of Computation

Author Biography

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.

Table of Contents

Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration
The Arithmetic-Geometric Mean Iteration
1(4)
Gauss's Derivation of the Fundamental Limit Formula
5(2)
Basic Properties of Complete Elliptic Integrals
7(4)
Quadratic Transformations and Iterations and a Third Proof of the Fundamental Limit Formula
11(7)
Jacobi's Differential Equation and a Fourth Proof of the Fundamental Limit Theorem
18(5)
Legendre's Relation
23(5)
Elliptic Functions
28(5)
Theta Functions and the Arithmetic-Geometric Mean Iteration
A Theta Series Solution to the AGM
33(3)
Poisson Summation
36(4)
Poisson Summation and the AGM
40(4)
The Derived Iterationa and Some Convergence Results
44(2)
Two Algorithms for π
46(6)
General Theta Functions
52(5)
The Landen Transformation
57(5)
Jacobi's Triple-Product and Some Number Theoretic Applications
Jacobi's Triple-Product Identity
62(5)
Some Further Theta Function Identities
67(9)
A Combinatorial Approach to the Triple-Product Identity
76(2)
Bressoud's ``Easy Proof'' of the Rogers-Ramanujan Identities
78(3)
Some Number Theoretic Applications
81(6)
The Mellin Transform and the Zeta Function
87(4)
Evaluation of Sums of Reciprocals of Fibonacci Sequences
91(11)
Higher Order Transformations
102(50)
A First Approach to Higher Order Transformations
102(7)
An Elementary Transcendental Approach to Higher Order Transformations
109(3)
Elliptic Modular Functions
112(7)
The Modular Equations for λ and j
119(7)
The Modular Equation in u-v Form
126(10)
The Multiplier
136(6)
Cubic Modular Identities
142(10)
Modular Equations and Algebraic Approximations to π
152(48)
Singular Values of the Second Kind
152(3)
Calculation of α
155(9)
Further Formulae for α
164(5)
Recursive Approximation to π
169(8)
Generalized Elliptic Integrals and Rational and Algebraic Series for 1/π and 1/K
177(14)
Other Approximations
191(9)
The Complexity of Algebraic Functions
200(19)
Complexity Concerns
200(4)
The Fast Fourier Transform (FFT)
204(5)
Fast Multiplication
209(3)
Newton's Method and the Complexity of Algebraic Functions
212(7)
Algorithms for the Elementary Functions
219(11)
π and Log
219(5)
Theta Function Algorithms for Log
224(2)
The Complexity of Elementary and Elliptic Functions
226(4)
General Means and Iterations
230(51)
Abstract Means
230(9)
Equivalence of Means
239(4)
Compound Means
243(6)
Convergence Rates and Some Examples
249(7)
Carlson's Integrals and More Examples
256(7)
Series Expansions of Certain Means
263(3)
Multidimensional Means and Iterations
266(7)
Algebraic Iterations and Functional Relations
273(8)
Some Additional Applications
281(35)
Sums of Two Squares
281(7)
(Chemical) Lattice Sums
288(13)
Odd-Dimensional Sums and Benson's Formula
301(5)
The Quintuple-Product Identity
306(3)
Quintic and Septic Multipliers and Iterations
309(7)
Other Approaches to the Elementary Functions
316(21)
Classical Approximations
316(10)
Reduced Complexity Methods
326(11)
Pi
337(50)
On the History of the Calculation of π
337(10)
On the Transcendence of π
347(15)
Irrationality Measures
362(25)
Bibliography 387(9)
Symbol List 396(9)
Index 405

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