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Borwein J., Bailey D., Girgensohn R. — Experimentation in Mathematics: Computational Paths to Discovery
Borwein J., Bailey D., Girgensohn R. — Experimentation in Mathematics: Computational Paths to Discovery



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Название: Experimentation in Mathematics: Computational Paths to Discovery

Авторы: Borwein J., Bailey D., Girgensohn R.

Аннотация:

Sometimes mathematicians like to believe that theorems spring full- fledged from their brains, given birth solely by the power of naked mind. Borwein, Bailey, and Girgensohn take a different approach, using computer programs like Maple and Mathematica to explore hypothesis and generate ideas. Their approach is still rigorous, however, because the insight gained from computer experimentation is then incorporated into a rigorous proof. Topics are drawn primarily from analysis and number theory, including sequences and series, fourier series, zeta functions, partitions and powers, and primes and polynomials. Advanced undergraduate students with solid experience in these areas, or beginning graduate students should find this book accessible; in either case, no programming knowledge is assumed. The first volume of this work is Mathematics by Experiment: Plausible Reasoning in the 21st Century; each of the two can stand on its own.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 368

Добавлена в каталог: 11.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abel summation      87 111
Abel, Niels Henrik      16 50
Adamchik, Victor      ix
Additive partitions      see partitions
Additive sequences      245
Affine variety      see variety
Affleck, Ian      36
AGM iteration, complex      213
Agoh's conjecture      232
Agrawal, Manindra      302
Ahmed, Zafar      17
Airy, George, Airy function      61
Airy, George, Airy integral      60
Aliquot sequences      246
Amicable numbers      245—246
Andrews' convolution      189 209
Apery, Roger      136
Archimedes      3 285
Archimedes' inequality      2$\ddag$
Argument principle      274
Arithmetic-geometric mean (AGM)      190 263 265
Arithmetic-geometric mean (AGM), AG2      288
Arithmetic-geometric mean (AGM), generalized      263—268
Aronson's sequence      58
Artjuhov, Mikhail      301
Banach contraction theorem      270—271
Barrier functions      254—256
Bauschke, Heinz      ix
Bbounded variation      74
BBP-type formulas      49 315 317
Beatty's theorem      59
Bell numbers      41 58
Bell numbers, Bell polynomials      58
Bender, Carl      323
Benson's formula      208
Bernoulli convolution      92—98
Bernoulli convolution, Bernoulli measure      92
Bernoulli numbers      29 133 135 136 232 310 315 323 325
Bernoulli numbers, continued fractions      324
Bernoulli, Daniel      71
Bernstein, Daniel      303
Bernstein, Sergi, Bernstein operators      102
Bernstein, Sergi, Bernstein polynomials      127—128
Berry, Michael      1$\ddag$ 42
Bertrand's postulate      171
Bescovitch, A.S.      260
Bessel functions      34 118 167 200
Bessel functions, modified      199
beta function      123
Beta function, derivatives      145—146
Beta function, digamma representation      146
Beta function, Riemann zeta function      169
Beukers, Frits      3
Bezier curves      127
Bezout's theorem      240
Binet's mean value theorem      122
Binomial series      287
Binomial series, evaluating      20—28
Bohr, Harald      291
Boltzmann — Shannon entropy      272 274
Bolzano, Bernhard      109
Boole summation formula      29 317
Borel. Emile      66
Borwein, Peter      ix
Bradley's formula      317
Bradley, David      ix
Broadhurst, David      49 144 146 161
Brouncker, William      30
Brouwer, Luitzen Egbertus Jan      295—297
Brouwer, Luitzen Egbertus Jan, Brouwer's fixed point theorem      295
Buchberger's algorithm      236
Calderbank, A.R.      333
Canfield, E.Rodney      41
Cantor, Georg      71
Cantor, Georg, Cantor function      93
Caratheodory, Constantin      183
Carleman's inequality      284
Carmichael numbers      226 227
Cassaigne, Julien      2 15
Castlejau's algorithm      127
Catalan, Eugene, Catalan numbers      53 210
Catalan, Eugene, Catalan zeta function      140 197
Catalan, Eugene, Catalan's constant      20 28 106 169 197 306 315 326
Catalan, Eugene, evaluating      316
Cauchy, Augustin-Louis, Cauchy convolution      180
Cauchy, Augustin-Louis, Cauchy — Lindeloef theorem      216
Cayley, Arthur      189
Cayley, Arthur, Cayley — Hamilton theorem      205
Cayley, Arthur, Cayley — Menger determinant      233 234
Cesaro summation      48 75 87 111
Chaitin, Gregory      ix 334 335f
Champernowne number      47
Chandrasekharan, K.      112
Chang, Jen      ix
Chebvshev polynomial      15
Cheyvshev, Pafnuty      50
Chudnovsky, David and Gregory      ix 321
Clausen, Thomas      136
Clausen, Thomas, Clausen functions      27
Clausen, Thomas, Clausen's product      50
Clausen, Thomas, multiple Clausen values      162
Cohen, Henri      246
Computer algebra packages, bugs      98
Computer algebra packages, integral evaluation      17—20
Computer algebra packages, product evaluation      1 7
Conjugate function      see Fenchel — Young conjugate
Continued fractions      11—15 28—36
Continued fractions, hypergeometric functions      31
Continued fractions, numerical accuracy      11
Contraction map      270
Convergence to arctan      8$\ddag$
Convergence to measures      96 src='/math_tex/81a7d1690f377d0f98d5387858d71ac282.gif'
Corless, Robert      ix
Crandll, Richard      ix 233 300 311
Crandll, Richard, Madelung's constant      200 208
D'Alembert, Jean      70
DAD problems      272 293—295
Dauberhies. Ingrid      93
David Parker's DPGraph      257
De La Vallee Poussin      244
Dedekind $\eta$-function      187
Deligne words      162
Devlin, Keith      184$\dag$
Diagonalization algorithm      273
Difference equations      92
Digamma function      64
Dijk paths      210
Dilogarithms      49
Dirichlet, Lejeune      71 244
Dirichlet, Lejeune, Dirichlet class number      142
Dirichlet, Lejeune, Dirichlet kernel      104 108
Dirichlet, Lejeune, Dirichlet operator      104
Dirichlet, Lejeune, Dirichlet series      141
Dirichlet, Lejeune, primitive L-series      140 171
Doubly stochastic matrix      272 293
Doubly stochastic pattern      272
Drinfeld — Deligne conjecture      144
Dunham, William      67
Dyson, Freeman      131$\dag$
Ehrhardt polynomial      258—259
Eigenvector      271
Einstein, Albert      89
Ek functions      290$\ddag$
Ekeland variational principle      269
Ekeland variational principle, Banach contraction      270 271
Elkies, Noam D.      172
Elliptic integrals      117 118 190 284—285
Elliptic integrals, mean iteration      282
Eratosthenes, sieve of Eratosthenes      301
Erdoes, Paul      94 138 188
Error function      105
Euclidean algorithm      305
Euler, Leonhard      67
Euler, Leonhard, Basel problem      67
Euler, Leonhard, continued fractions      30 35
Euler, Leonhard, Euler function      326
Euler, Leonhard, Euler numbers      29 140
Euler, Leonhard, Euler polynomials      326
Euler, Leonhard, Euler product      134
Euler, Leonhard, Euler sums      142 144
Euler, Leonhard, Euler sums, double Euler sums      149 152 203
Euler, Leonhard, Euler sums, unit Euler sum      153
Euler, Leonhard, Euler — Maclaurm summation formula      29 167 309—310 312 315
Euler, Leonhard, Euler — Maclaurm summation formula, infinite series      315—317
Euler, Leonhard, Euler — Maclaurm summation formula, MZVs      317—318
Euler, Leonhard, Euler's beta integral      282
Euler, Leonhard, Euler's integral      285
Euler, Leonhard, Euler's pentagonal number theorem      185
Euler, Leonhard, Euler's totient function      54 248
Euler, Leonhard, zeta function      134 135
Eureka (journal)      3
Exponentiation function      219 219 220*
Faber — Schauder basis      see Schauder basis
Faber, Georg      114
Face-centered cubic (FCC)      119
Fast Fourier transform (FFT)      85 303 305
Fast Fourier transform (FFT), convolutions      85
Fateman, Richard      ix
Fatou's condition      88
Fee, Greg      ix
Fejer's series      90
Fejer, Lipot, Fejer operator      104
Fejer, Lipot, Fejer's sum      85
Fejer, Lipot, Fejer's theorem      84
Fenchel, Werner, Fenchel duality      121
Fenchel, Werner, Fenchel — Young conjugate Lambert W function      280
Ferguson, Helaman      ix
Fermat primes      232
Fermat's little theorem      225
Fermat, Pierre de      245
Ferrer diagram      184$\ddag$
Feynman, Richard      299$\dag$
Fibonacci numbers      215
Fibonacci sequence      200 203 222
Fibonacci, Leonardo      200$\dag$ 222—223
Finch, Steven      ix 245
Fixed point      270
Fixed point, Banach contraction theorem      270
Fourier series      72 72—77
Fourier series, approximate identity      85
Fourier series, Bernoulli convolution      92—98
Fourier series, convergence      74
Fourier series, convergence, $L^1(T)$      75
Fourier series, convergence, pointwise      74 75
Fourier series, convergence, uniform      74
Fourier series, convolution theorems      83—85
Fourier series, Hamburger moment problem      91
Fourier series, Jordan test      74 89
Fourier series, kernels, $(C, \alpha)$-kernel      112
Fourier series, kernels, Blackmail kernel      113
Fourier series, kernels, de la Vallee-Poussin kernel      112
Fourier series, kernels, definition      85
Fourier series, kernels, Dirichlet kernel      87 104 108
Fourier series, kernels, Fejer kernel      75 78 87 104 111
Fourier series, kernels, Fejer — Korovkin kernel      113 128
Fourier series, kernels, Gauss kernel      78
Fourier series, kernels, Jackson kernel      113 128
Fourier series, kernels, Parseval's equation      76
Fourier series, kernels, Poisson kernel      128
Fourier series, pathologies      107
Fourier series, periodization      81
Fourier series, Poisson summation formula      81
Fourier transform      78 78—80
Fourier transform, $L^1(R)$      78
Fourier transform, $L^2(R)$      79 79
Fourier transform, Jordan's theorem      80
Fourier transform, Riemann — Lebesgue lemma      78
Fourier transform, uniqueness theorem      78
Fourier, Joseph      69$\dag$ 70—72
Fubini's theorem      18 99 105
Fundamental theorem of algebra      296-297
Galway, William      ix
Gamma function      67 117 128 197 287—288
Gamma function, $AG_2$      288
Gamma function, elliptic integrals      193
Gamma function, identities      165
Gamma function, incomplete      326
Gamma function, infinite products      5—7
Gamma function, recycling      326
Gamma function, Riemann zeta function      132
Gamma function, Selberg integral      282
Gamma function, sine integral      122
Gamma function, Zagier conjecture      160
Garsia, A.      95
Gauss, Carl Friedrich      200
Gauss, Carl Friedrich, arithmetic-geometric mean      265
Gauss, Carl Friedrich, continued fractions      31 35
Gaussian quadrature      307
Gersgorin circles      215
Gibbs, J.Willard, Gibbs phenomenon      89$\dag$ 90 89—90
Giuga, G.      225
Giuga, G., Giuga number      226
Giuga, G., Giuga numbers      227
Giuga, G., Giuga sequence      231 231 232
Giuga, G., Giuga sequences      246
Giuga, G., Giuga's prime number conjecture      225
Goedel, Kurt      332
Goldston, Daniel      259
Google      41
Gosper, Bill      62 138
Graham, Ronald      ix
Granville, Andrew      ix 260
Grassman manifolds      242
Gregory, James, Gregory series      326
Gregory, James, Gregory's series      28
Grelling's paradox of heterologicality      332
Groebner basis      233 236—240
Groebner basis, grevlex basis      237
Haar basis      113
Hadamard inequality      56
Hadamard, Jacques      295
Haenel, Christoph      ix
Halley, Edmond      323
Hamburger moment problem      91
Hankel, Hermann, Hankel loop integral      288
Hankel, Hermann, Hankel matrix      250
Hardy, G.H.      109 187 193 198 259 260
Hata, Masayoshi      4
Herivel, John      69
Heron's formula      235
Heterological property      332
Hida, Yozo      ix
Hilbert matrix      250
Hilbert's theorem      258
Hilbert, David      vii$\dag$ 133
Hjab, Omar      37
Hoch and Stern information measure      273
Hoelder convolution      156—157
Homological property      332
Hooke and Jeeves algorithm      see pattern search algorithm
Horner's rule      304
Hurwitz, Adolf      249
Hyperbolic polynomials      254
Hyperbolicity cone      257$\ddag$
Hypergeometric function      35 50 117 211 285
Hypergeometric function, arithmetic-geometric mean      265 266
Hypergeometric function, contiguity relation      31
Hypergeometric function, continued fractions      31—35
Hypergeometric function, Euler's integral      285
Hypergeometric function, Gamma function      286
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