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

by Borwein, Jonathan M.; Borwein, Peter B.

• ISBN13: 9780471315155
• ISBN10: 047131515X
• Edition: 1st
• Format: Paperback
• Copyright: 1998-07-13
• Publisher: Wiley-Interscience

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