正版 π与算术几何平均:关于解析数论和计算复杂性的研究(英文

Chapter 1 Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration
1.1 The Arithmetic-Geometric Mean Iteration
1.2 Gausss Derivation of the Fundamental Limit Formula
1.3 Basic Properties of Complete Elliptic Integrals
1.4 Quadratic Transformations and Iterations and a Third Proof of the Fundamental Limit Formula
1.5 Jacobis Differential Equation and a Fourth Proof of the Fundamental Limit Theorem
1.6 Legendres Relation
1.7 Elliptic Functions
Chapter 2 Theta Functions and the Arithmetic-Geometric Mean Iteration
2.1 A Theta Series Solution to the AGM
2.2 Poisson Summation
2.3 Poisson Summation and the AGM
2.4 The Derived Iteration and Some Convergence Results
2.5 Two Algorithms for π
2.6 General Theta Functions
2.7 The Landen Transformation
Chapter 3 Jacobis Triple.Product and Some Number Theoretic Applications
3.1 Jacobis Triple-Product Identity
3.2 Some Further Theta Function Identities
3.3 A Combinatorial Approach to the Triple-Product Identity
3.4 Bressouds \"Easy Proof\" of the Rogers-Ramanujan Identities
3.5 Some Number Theoretic Applications
3.6 The MeUin Transform and the Zeta Function
3.7 Evaluation of Sums of Reciprocals of Fibonacci Sequences
Chapter 4 Higher Order Transformations
4.1 A First Approach to Higher Order Transformations
4.2 An Elementary Transcendental Approach to Higher Order Transformations
4.3 Elliptic Modular Functions
4.4 The Modular Equations for A and j
4.5 The Modular Equation in u-v Form
4.6 The Multiplier
4.7 Cubic Modular Identities
Chapter 5 Modular Equations and Algebraic Approximations to π
5.1 Singular Values of the Second Kind
5.2 Calculation of a
5.3 Further Formulae for α
5.4 Recursive Approximation to π
5.5 Generalized Elliptic Integrals and Rational and Algebraic Series for 1/π and 1/K
5.6 Other Approximations
Chapter 6 The Complexity of Algebraic Functions
6.1 Complexity Concerns
6.2 The Fast Fourier Transform (FFT)
6.3 Fast Multiplication
6.4 Newtons Method and the Complexity of Algebraic Functions
Chapter 7 Algorithms for the Elementary Functions
7.1 π and Log
7.2 Theta Function Algorithms for Log
7.3 The Complexity of Elementary and Elliptic Functions
Chapter 8 General Means and Iterations
8.1 Abstract Means
8.2 Equivalence of Means
8.3 Compound Means
8.4 Convergence Rates and Some Examples
8.5 Carlsons Integrals and More Examples
8.6 Series Expansions of Certain Means
8.7 Multidimensional Means and Iterations
8.8 Algebraic Iterations and Functional Relations
Chapter 9 Some Additional Applications
9.1 Sums of Two Squares
9.2 (Chemical) Lattice Sums
9.3 Odd-Dimensional Sums and Bensons Formula
9.4 The Quintuple-Product Identity
9.5 Quintic and Septic Multipliers and Iterations
Chapter 10 Other Approaches to the Elementary Functions
10.1 Classical Approximations
10.2 Reduced Complexity Methods
Chapter 11 Pi
11.1 On the History of the Calculation of π
11.2 On the Transcendence of π
11.3 Irrationality Measures
Bibliography
Symbol List
Index

本书主要包括完全椭圆积分
和算术几何平均迭代次数、
算术几何平均迭代、雅可比
三重积及其一些数论应用、
高阶转换、模方程和代数近
似值、代数函数的复杂性、
初等函数的算法、一般方法
及迭代、平方和的应用、经
典近似、简化复杂性方法等
内容。其具体内容如下：第
一章，完全椭圆积分与算术
几何平均迭代；第2章，算
术几何平均迭代；第3章，
雅克比三重积及其一些数值
理论应用；第4章，高阶转
换；第5章，模方程和代数
近似；第6章，代数函数的
复杂性；第7章，初等函数
的算法；第8章，常规方法
与迭代；第9章，一些其他
应用；第10章，处理初等函
数的其他方法。本书适合于
参加数学竞赛的选手以及数
学爱好者参考使用。