Chapter 1 Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration 1.1 The Arithmetic-Geometric Mean Iteration 1.2 Gauss's 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 Jacobi's Differential Equation and a Fourth Proof of the Fundamental Limit Theorem 1.6 Legendre's 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 Jacobi's Triple.Product and Some Number Theoretic Applications 3.1 Jacobi's Triple-Product Identity 3.2 Some Further Theta Function Identities 3.3 A Combinatorial Approach to the Triple-Product Identity 3.4 Bressoud's "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 Newton's 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 Carlson's 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 Benson's 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
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