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Borwein J.M., Borwein P.B. — Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
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Название: Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
Авторы: Borwein J.M., Borwein P.B.
Аннотация: Presents new research revealing the interplay between classical analysis and modern computation and complexity theory. Two intimately interwoven threads run though the text: the arithmetic-geometric mean (AGM) iteration of Gauss, Lagrange, and Legendre and the calculation of pi[l.c. Greek letter]. These two threads are carried in three directions. The first leads to 19th century analysis, in particular, the transformation theory of elliptic integrals, which necessitates a brief discussion of such topics as elliptic integrals and functions, theta functions, and modular functions. The second takes the reader into the domain of analytic complexity — Just how intrinsically difficult is it to calculate algebraic functions, elementary functions and constants, and the familiar functions of mathematical physics? The answers are surprising, for the familiar methods are often far from optimal. The third direction leads through applications and ancillary material — particularly the rich interconnections between the function theory and the number theory. Included are Rogers-Ramanujan identities, algebraic series for pi[l.c. Greek letter], results on sums of two and four squares, the transcendence of pi[l.c. Greek letter] and e[ital.], and a discussion of Madelung's constant, lattice sums, and elliptic invariants. Exercises.
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Рубрика: Математика /
Статус предметного указателя: Готов указатель с номерами страниц
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Год издания: 1987
Количество страниц: 414
Добавлена в каталог: 11.04.2008
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Предметный указатель
, quadratic algorithm for 50
, irrationality of 366
, irrationality of 369
, series for 190 379
Abel 5 28
Abramowitz and Stegun (64) 15 321
Adams (66) 371
Aho, Hopcroft and Ullman (74) 201 203 206 211
Aiazy et al. (85) 269 272
Algebraic addition theorem 29 30 32
Algebraic function 273 276
Algebraic functional relation 273
Algebraic integrals 5
Algebraic number 351 352
Algebraic series for and 1/K 181—190
Algebraic transformation 273 274 278
Algebraic transformation group 273
Alladi and Robinson (79) 371
Almqvist (Pr) 99
Almqvist and Berndt (Pr) 3 196 371
Alpha, singular value of second kind 152
Alpha, singular value of second kind, cubic 160
Alpha, singular value of second kind, monotonicity 153
Alpha, singular value of second kind, quadratic 158
Alpha, singular value of second kind, quart ic 161
Alpha, singular value of second kind, quintic 310
Alpha, singular value of second kind, recursions 156
Alpha, singular value of second kind, septic 311
Alpha, singular value of second kind, theta function form 154
Alternating Series Test 291
Andrews (76) 67 86
Andrews (86) 286
Apery 365 371 379
Apostol (74) 37 40
Apostol, (76a) 86
Apostol, (76b) 116
Archimedean mean iterations 246 249 251 252
Archimedean means 259
Archimedean product 246 249 250 253 262
Archimedes 243 249 337 338
Archimedes' method 249 338
Arclemniscate 25
Arclemniscate sine 259
Aristophanes 347
Arithmetic-geometric mean (AGM) 1
Arithmetic-geometric mean (AGM), AGM relation 1
Arithmetic-geometric mean (AGM), AGM relation, complex starting values 15
Arithmetic-geometric mean (AGM), calculation of 341
Arithmetic-geometric mean (AGM), computational performance 330
Arithmetic-geometric mean (AGM), continued fraction 188
Arithmetic-geometric mean (AGM), Gaussian 181 241 257
Arithmetic-geometric mean (AGM), Jacobi's identity 35
Arithmetic-geometric mean (AGM), Legendre form 3 45
Arithmetic-geometric mean (AGM), matrix AGM 223 226
Arithmetic-geometric mean (AGM), multidimensional AGM 272
Arithmetic-geometric mean (AGM), quartic AGM 17 51 254
Arithmetic-geometric mean (AGM), theta forms 146
Arithmetic-geometric mean (AGM), variant of 254
Askey (80) 89 308
Backstrom (81) 98 99
Bailey, D. (88) 211 341 342
Bailey, W. 181
Bailey, W. (35) 184
Baker 323 348
Baker (75) 349 351 354 356 363 371
Baker and Graves-Morris (81) 320 323
Baltantine (39) 347
Baxter 80
Beckenbach (50) 236 269
Beckmann 346
Beckmann (77) 342
Beeler et al. (72) 220 222
Bell (27) 36
Bellman (61) 39 55 56 86 89
Bemdt (Pr) 142 150 177 196 309
Benson 301 302 303
Benson's formula 301 303
Bernoulli numbers 383
Bernoulli, Jacques 340
Bernoulli, Jean 340
Bessel function transform 39
Best approximants, existence 321
Best approximants, uniform polynomial 316
Best approximants, uniform rational 317
Best approximation property 373
beta function 24 88
Beukers 195
Beukers (79) 365
Bhargava and Chandrashekar Adiga (84) 81
Biagioli (Pr) 146
Birkhoff (73) 10 18 116
Birkhoff and Rota (69) 9
Bit complexity 200
Bohr 89
Borchardt 250 256 257 263
Borchardt (1888) 272
Borchardt's algorithm 250 256 263
Borodin and Munro (75) 206 208
Borwein and Borwein (84a) 44 50 51 222 223
Borwein and Borwein (84b) 108 174
Borwein and Borwein (84c) 174
Borwein and Borwein (84d) 222 225
Borwein and Borwein (86) 174
Borwein and Borwein (Pr) 313
Borwein, Borwein and Taylor (85) 290 291 292
Borwein, D. and Borwein, J. (86) 291
Borwein, J.M. (85) 163 164
Borwein, P.B. (85) 218
Borwein, P.B. (Pr) 254
Bouyer 341
Bowman (53) 29
Boyer (68) 347
Braess (84) 325
Brent 51
Brent (76a) 48 222 229
Brent (76b) 222
Brent (76c) 213 216 217 222 329 330
Brent and McMillan (80) 336
Bressoud 78
Bressoud (83) 77
Bring 136
Bromwich (26) 189 291 344
Brouncker 338 345
Brouncker's continued fraction 339 345
Bundschuh (71) 371
Buslaev, Gonchar and Suetin (84) 323
Calculating powers of x 208
Carlitz (71) 71
Carlson 248 257 261
Carlson (71) 7 256
Carlson (75) 261
Carlson (78) 262
Carlson's integrals 256
Carlson's log 248 257
Catalan 66
Catalan's constant 198 371 386
Cauchy 76 77 292
Cauchy's binomial theorem 76
Cayley (1874) 106 133 138 140 142
Cayley (1895) 22 102 138
Chandrasekharan (85) 116
Cheney (66) 321
Chudnovsky and Chudnovsky (84) 368 371
Clark (71) 349
Class number 141 295
Clausen 179 188 189 190
Clausen's hypergeometric product 179 188 189 190
Cohen and Nussbaum (87) 273
Comparable means 244 246 247
Complement 3 178
Complementary integral 8
Complementary modulus 8
Complete elliptic integrals 7
Complete elliptic integrals of first kind (K) 7
Complete elliptic integrals of second kind 7
Complete elliptic integrals, complementary integrals 8
Complete elliptic integrals, cubic algorithm for K 107
Complete elliptic integrals, differential equations 9
Complete elliptic integrals, E in terms of theta functions 42
Complete elliptic integrals, higher order transformations 102
Complete elliptic integrals, homogeneous forms of K and E 12
Complete elliptic integrals, K in terms of gamma function 297 298
Complete elliptic integrals, K in terms of theta functions 35
Complete elliptic integrals, moments of K and 198
Complete elliptic integrals, quadratic transformations 12
Complete elliptic integrals, quadratically convergent algorithms 14
Complete elliptic integrals, series expansions of 8
Complexity of algebraic functions 215
Complexity of theta computation 95
Complexity, 219
Complexity, bit 200
Complexity, elementary functions 226
Complexity, elliptic integral calculations 227
Complexity, Jacobian elliptic functions 227
Complexity, Lambert series 98
Complexity, log 219
Complexity, lower bound for log and exp 227
Complexity, operational 201
Complexity, transcendental functions 219 226
Compound of means 243
Comtet 386
Conjugate divisors 297
Constructibility 347 349—351
Continued fraction 372—377
Contra-harmonic mean 255
Convergents 372
Cook 69 213
Cook and Aanderaa (69) 69
Cooley and Tuckey (65) 65
Cooley, Lewis and Welch (67) 65
Cox (85) 15
Cubic invariants 173
Cubic modular identities 142
Cubic recursions for Ramanujah's invariants 145
Dase 340
Davenport (81) 226
Davis (79) 365 371
Denninger (84) 89
Diagonal mapping 230
Dickson (29) 293
Dickson (71) 55 66 85 141 284 287 292
Dilogarithm 381
Dirichlet 86
Dirichlet class number formulae 294 295
Dirichlet L functions 289
Discrete Fourier Transform 204
Discriminants 294
Disjoint discriminants 293
Domination 239
DuVal (73) 23 30 133
e, irrationality of 352
e, transcendence of 353 359
Eagle (58) 29
Edwards (79) 252
Elementary symmetric mean function 270
Elementary symmetric polynomial 356
Ellipse, arc length of 8
Elliptic functions 29 30 31
Elliptic functions order 30
Elliptic functions, cn 29
Elliptic functions, degenerate elliptic functions 29
Elliptic functions, dn 29
Elliptic functions, fundamental parallelogram 29 53
Elliptic functions, Jacobian 29
Elliptic functions, lattice 29
Elliptic functions, odd elliptic functions 31
Elliptic functions, period of 29
Elliptic functions, sn 29 57
Elliptic integrals see “Complete and Incomplete”
Epstein 303
Equivalence of operations 213
Equivalent iterations 252
Equivalent mean 231 239
Erdeiyi et al. (53) 10 30 55 178 179 181
Estimates of e 363
Eta-multiplier 311 314
Eta-multiplier of order p 311
Euler 3 18 64 67 77 189 264 286 306 307 340 344 346 348 381 382 383
Euler numbers 383
Euler numbers, computation of 336
Euler — Mascheroni constant 336
Euler's addition theorem 18
Euler's formula 189
Euler's identity 64
Euler's pentagonal number theorem 64 66 80 306
Euler's totient function 301
Evaluation problem 204
Ewell (81) 66
Ewell (82) 306
Ewell (83) 86
Ewell (86) 151
Exponential, algorithms for 227
Exponential, approximations to 317 318 320 321 322 323 327 330 331 335
Exponential, series for 320
Factorial calculation 218
Fast base conversion 218
Fast computation 50
Fast Fourier transform (FFT) 204 206 211 212
Fast matrix multiplication 202
Fast multiplication 209
Fast polynomial division 207
Fast polynomial evaluation 208
Fast polynomial multiplication 206
Feldman 371
Felton 341
Ferguson 340
Fermat 82 286
Fibonacci numbers 94 151 287 375
Fibonacci sequences and series 91 97 98 100
Finite Fourier transform 204
Foster and Phillips (84a) 260
Foster and Phillips (84b) 247 252
Fully monotone 291
Fundamental limit theorem 5
Fundamental limit theorem, first proof 5
Fundamental limit theorem, fourth proof 21
Fundamental limit theorem, second proof 6
Fundamental limit theorem, third proof 12
Fundamental regions 113
Fundamental sets 113
Fundamental unit 294
Galois group for 123
Galois group for , and 136
Gamma function 24 27 28 87 332 336
Gamma function, duplication formula 88 90
Gamma function, evaluation of K and E 25 27 189 191 297 298
Gammel 323
Gauss 1 3 5 8 10 28 43 51 85 86 250 256 284 286
Gauss (1866) 5 7 35 44 48 52 65
Gauss's multiplication formula 90
Gaussian binomial coefficients 76
Gaussian mean iterations 246 249 251 252
Gaussian product 246 248 253
Gaussian sums 83 86
Gelfond 348
Gelfond — Schneider theorem 277 278 348
Genera 295
Generalized complete elliptic integrals of first kind 178
Generalized complete elliptic integrals of second kind 178
Generalized complete elliptic integrals transformations 179 180 185
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