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Borwein J., Bailey D., Girgensohn R. — Experimentation in Mathematics: Computational Paths to Discovery
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Название: Experimentation in Mathematics: Computational Paths to DiscoveryАвторы: 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: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2004Количество страниц: 368Добавлена в каталог: 11.06.2008Операции: Положить на полку |
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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 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 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 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 187 Deligne words 162 Devlin, Keith 184 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 Ehrhardt polynomial 258—259 Eigenvector 271 Einstein, Albert 89 Ek functions 290 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 Feynman, Richard 299 Fibonacci numbers 215 Fibonacci sequence 200 203 222 Fibonacci, Leonardo 200 222—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, 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, 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, 78 Fourier transform, 79 79 Fourier transform, Jordan's theorem 80 Fourier transform, Riemann — Lebesgue lemma 78 Fourier transform, uniqueness theorem 78 Fourier, Joseph 69 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, 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 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 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 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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