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Borwein J.M., Borwein P.B. — Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
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.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1987

Количество страниц: 414

Добавлена в каталог: 11.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\mathrm{Exp}(\pi)$, quadratic algorithm for      50
$\zeta(2)$, irrationality of      366
$\zeta(3)$, irrationality of      369
$\zeta(3)$, 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 $1/\pi$ 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 $\pi$      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, $\pi$      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 $\mathrm{F}_\mathrm{p}$      123
Galois group for $\mathrm{W}_5$, and $\mathrm{W}_7$      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
1 2 3
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