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