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Young R.M. — Excursions in Calculus: An Interplay of the Continuous and the Discrete
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Íàçâàíèå: Excursions in Calculus: An Interplay of the Continuous and the DiscreteÀâòîð: Young R.M. Àííîòàöèÿ: he purpose of this book is to explore the rich and elegant interplay that exists between the two main currents of mathematics, the continuous and the discrete. Such fundamental notions in discrete mathematics as induction, recursion, combinatorics, number theory, discrete probability, and the algorithmic point of view as a unifying principle are continually explored as they interact with traditional calculus.
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Ãîä èçäàíèÿ: 1992Êîëè÷åñòâî ñòðàíèö: 408Äîáàâëåíà â êàòàëîã: 13.04.2008Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
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Ïðåäìåòíûé óêàçàòåëü
29 93 144 209 213 221 231—238 240 243 250 264 267 270 314 330—332 337 346 349 351—353 3x+1 problem 13 Abel, N.H. 224 231 304 315 357 Absolute convergence 227 Adams, J.A. 285 Adrien, R. 209 AGM see “Arithmetic-geometric mean” Aitken, R.G. 9 Algorithm Archimedean 243 Algorithm arithmetic-geometric mean 232—233 Algorithm Babylonian 192 Algorithm euclidean 26—29 35 128—130 144 Algorithm Fibonacci search 138 Alternating series 314 330 333—334 Amicable numbers 28 Andrews, G.E. 357 Angle, trisection of 310 328 Apostol, T.M. 348 Arago, M. 354 Archimedes 83 191 310 312 Archimedes Measurement of a Circle 244 Archimedes On Spirals 83 Archimedes Quadrature of the Parabola 310 Arithmetic mean 20 186—195 Arithmetic-geometric mean 231—238 Arithmetical function 219 Arithmetical triangle see “Pascal's triangle” Asymptotically equal 244 267 302 305 Average see also “Expected value; Function; Mean” 185—245 Axiom of Archimedes 312 Ayoub, R. 352 Bachet's theorem 75 Ball, W.W.R. 24 Barrow, I. 344 Barsky, B.A. 285 Bartels, R.H. 285 Bartok, B. 124 Beatty, J.C. 285 Beatty, S. 156 Bell numbers 167 Bell, E.X. 100 268 Bernoulli numbers 87—94 99 160—161 352 355 Bernoulli polynomials 87—93 99 Bernoulli's inequality 22 Bernoulli, James 78 148 153 231 338 345 Bernoulli, James Ars conjectandi 78 86 Bernoulli, John 231 338 345 Bernoulli, N. 32 160 Bernstein polynomials 283—284 Bernstein, S.N. 275 282 Bertrand's paradox 244 Bertrand's postulate 34 71 306—307 Bezier Spline 284—286 Binet's formula 130 154 160 Binet, J.P.M. 130 Binomial coefficients 20—21 95 98—99 135 283—284 335—336 351 Binomial functions 65—68 Binomial series 255—259 Binomial theorem 21 216 Blumenthal, L.M. 322 Boas, R.P. 10 281 292 345 Bolyai, J. 306 Bombelli, R. 156 Borwein, J.M. and P.B. 234 237 Boscovich's hypothesis 209 Boscovich, R.J. 209 Boyer, C.B. 85 258 311 Bremmer, H. 278 Brent, R.P. 101 237 Brocard's problem 14 Brouncker, W. 144 331 Brousseau, A. 126 Buffon's Needle Problem 230 242 Bussey, W.H. 15 Cantor function 332—333 Cantor set 319—321 323 336 Cantor, G. 319 Cardioid 173 Carmichael numbers 116 Carroll, L. 1 127 309 Cassini's identity 127 138 141 Cassini, J.D. 127 Casting out nines 371 Catalan numbers 259 Catalan's problem 260 Catalan, E.C. 115 137 259 Cauchy kernel 279 Cauchy — Schwarz inequality 202 Cauchy, A.L. 75 188 223 315 Cauchy, A.L. Cours d'analyse 223 Cavalieri's formula 251 255 Cavalieri, B. 83 343 Cayley, A. 172 Central limit theorem 214 Cesaro, E. 135 324 353 Chaitin, G.J. 335 Chambered nautilus 149 151 Channel capacity 133 Characteristic triangle 343 Chebyshev's inequality 204 Chebyshev, P.L. 306 Child, J.M. 339 343—344 Chiliad 304 Chinese remainder theorem 373—375 Chudnovsky, D.V. and G.V. 237 Church, A. 52 Circle arc length of 232 Circle lattice points inside 221—223 Circle quadrature of 251—258 310 314 329 Circle rational points on 44—46 55 Cissoid of Diocles 328 Cohen, G.L. 101 coin tossing 225—230 248 254 263 283 Complex (Gaussian) integers 112 Composite numbers 7 Congruent numbers 369—377 Construction, straightedge and compass 142 310 Continued fractions 143—144 156 331 Contraction 180—181 Convex function 199 Convolution 164 Cox, D.A. 236 Coxeter, H.S.M. 24 Cube, duplication of 187 310 328 Cubes, sum of consecutive 78—81 Cummings, E.E. 1 Curve Bezier spline 284—286 Curve circle 44—46 55 221—223 232 251—258 310 314 329 Curve cissoid of Diodes 328 Curve devil's staircase 332—333 Curve ellipse 263 310 Curve folium of Descartes 56 Curve Gaussian see “Curve normal” Curve hyperbola 196 245 289—290 310 Curve Koch 323—325 336 Curve lemniscate 231—232 Curve logarithmic spiral 148—153 Curve normal 213 327 Curve parabola 310—314 Curve snowflake see “Curve Koch” Curve spiral of Archimedes 148 311 328—329 D'Alembert, J. le Rond 114 Davenport, H. 112 Davis, P.J. 219 225 304 de Moivre, A. 130 213 266 De Morgan, A. 15 De Parville, M. 24 Delahaye, J.P. 50 Delta function, Dirac 276—281 Dense 44 46 158 density 291 295—300 Derangement 37 165 Descartes, R. 56 148 344 Devaney, R.L. 183 Devil's staircase 332—333 Dewdney, A.K. 175 Dickson, L.E. 8 28 42 47 100 Dieudonne, J. 180 Difference equation 26 162 Difference operator 72 355 Difference set 336 Dilogarithm 350 Diocles 328 Diophantine equation 50—53 Diophantus of Alexandria 41 76 Diophantus of Alexandria Arithmetica 41 111 Diophantus of Alexandria, Book on Polygonal Numbers 76 Dirac delta function 276—281 Dirac, P.A.M. 276 Dirac, P.A.M., The Principles of Quantum Mechanics 276 Dirichlet's theorem 68 70 109—110 Dirichlet, P.G.Lejeune 104 245 306 Divergent series 223—225 355—356 Divine proportion see “Golden ratio” Divisors, number of 244—245 Divisors, sum of see also “Pentagonal number theorem” 115 Dorrie, H. 260 Douady, A. 173 175 Dubner, H. 70 Dudley, U. 69 Dynamical system 170 Dyson, E. 318 e 22 33 37 194 204—205 209 213 239—240 243 267 296 Eddington, A.S. 153 Edwards, A.W.E. 78 Edwards, C.H. 196 256 258 Egyptian fractions 295 Elementary function 266 268 Eliot, T.S. 4 368 Elkies, N. 42 ellipse 231—232 263 310 Elliptic integral 231—232 263 Encke, J.F. 306 Equiangular spiral see “Logarithmic spiral” Eratosthenes, sieve of 12 36 299 Erdos, P. 112 295 Error function 207 Euclid 312 Euclid Elements 27 43 101 Euclidean algorithm 26—29 35 128—130 144 Eudoxus 27 Eudoxus principle of 312 319 Euler 36 117 Euler — Maclaurin summation formula 346 354 Euler's constant 245 Euler's trinomial 64 Euler, L. 29 32 42 62 64 71 75 93 101 117 160 195 209 223 231 260 287 301 331 345 347 350—351 357 Exhaustion, method of 85 311 Expected value 163 225—230 Exponential function and series 195—197 Exponential generating function 161 167 Factorial 265—267 Factorial powers 72 Fair coin 225 248 Faltings, G. 50 Farin, G.E. 285 Fatou, P. 172 Fejer, L. 275 Feller, W. 32 164 226 Fermat numbers 34 62 71 116 Fermat's great theorem 110 221 316—317 Fermat's last theorem 4 41 47 94 Fermat's little theorem 100—104 108 116—118 Fermat, P. de see also “Infinite descent” 55 75—76 114 330 Fibonacci (Leonardo of Pisa) 124 Fibonacci (Leonardo of Pisa) Liber abaci 124 Fibonacci numbers 35 71 123—135 138 154 Fibonacci search algorithm 138 Figurate numbers 74 342 Fixed point 172 180—181 Fixed Point Iteration 180 Foley, J.D. 285 Folium of Descartes 56 Formally undecidable propositions 51 Fractal 175 318 326—327 Freedman, D. 207 Frenicle de Bessy, B. 103 Fresnel integrals 273 Function 297 Function 115 358—367 Function arithmetical 219 Function average value of 219—220 239 Function binomial 65—68 Function Cantor 332—333 Function convex 199—200 Function d(n) 244 Function dilogarithm 350 Function Dirac delta 276—281 Function elementary 266 268 Function error 207—213 Function Euler 36 117 Function exponential generating 161 167 Function generating 159 163—165 167 169 259 261—263 Function logistic 181—184 7 Function peaking 274 278—280 Function r(n) 220—223 317 Function Riemann zeta 352—353 Functional equation 150—152 158 211 Fundamental Theorem of Arithmetic see also “Gaussian integers” 23 48 371 Fundamental theorem of calculus 355 Fundamental theorem of information theory 133 Gage, P. 61 Gallon, F. 185 214—216 Gallon, F. Natural Inheritance 185 214 Gambler's ruin 162—164 Gardner, M. 261 Gauss, C.F. 63 110 118 120 207 209 213 231 247 264 304 369 376 Gauss, C.F., Disquisitiones Arithmeticae 120 369 Gauss, C.F., Theoria Motus 207 Gaussian curve 213 327 Gaussian integers 112—113 Generating function 159 163—165 167 169 259 261—263 Geometric mean 20 186—195 Geometric series 310—324 Gilbert, W.S. 57 Gilbreath's conjecture 308 Girard, A. 111 Gleick, J. 322 Gnomon 81 Godel,K. 4 51 Goldbach's conjecture 5—6 58 Goldbach, C. 64 260 Golden ratio 140—153 159 171 Golden rectangle 144—148 Golden section see “Golden ratio” Gosper, C. 238 Grabiner, J.V. 223 Graham, R.L. 12 295 Grandi, G. 223 Greatest common divisor see also “Euclidean algorithm” 26 Gregory's series 316 331 Gregory, J. 316 344 Guy, R.K. 8 42 116 295 Hadamard, J. 306 Halmos, P.R. x Halting problem 51 Hardy, G.H. xi 5 224 369 Harmonic mean 187 201 Harmonic number 165 296 Harmonic series 287—288 295—296
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