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Young R.M. Ч Excursions in Calculus: An Interplay of the Continuous and the Discrete
Young R.M. Ч Excursions in Calculus: An Interplay of the Continuous and the Discrete

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Ќазвание: Excursions in Calculus: An Interplay of the Continuous and the Discrete

јвтор: Young R.M.

јннотаци€:

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.

As the topics and applications will show, much of the material has never been presented in this level. The book is addressed primarily to well-trained calculus students and those who teach them, but it can also serve as a supplement in a traditional calculus course for anyone who wants to see more. The problems, taken for the most part from probability, analysis, and number theory, are an integral part of the text. There are over 400 problems presented in this book.


язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1992

 оличество страниц: 408

ƒобавлена в каталог: 13.04.2008

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$\pi$      29 93 144 209 213 221 231Ч238 240 243 250 264 267 270 314 330Ч332 337 346 349 351Ч353
3x+1 problem      13
Abel, N.H.      224 231 304 315 357
Absolute convergence      227
Adams, J.A.      285
Adrien, R.      209
AGM      see УArithmetic-geometric meanФ
Aitken, R.G.      9
Algorithm Archimedean      243
Algorithm arithmetic-geometric mean      232Ч233
Algorithm Babylonian      192
Algorithm euclidean      26Ч29 35 128Ч130 144
Algorithm Fibonacci search      138
Alternating series      314 330 333Ч334
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 186Ч195
Arithmetic-geometric mean      231Ч238
Arithmetical function      219
Arithmetical triangle      see УPascal's triangleФ
Asymptotically equal      244 267 302 305
Average      see also УExpected value; Function; MeanФ 185Ч245
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      87Ч94 99 160Ч161 352 355
Bernoulli polynomials      87Ч93 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      283Ч284
Bernstein, S.N.      275 282
Bertrand's paradox      244
Bertrand's postulate      34 71 306Ч307
Bezier Spline      284Ч286
Binet's formula      130 154 160
Binet, J.P.M.      130
Binomial coefficients      20Ч21 95 98Ч99 135 283Ч284 335Ч336 351
Binomial functions      65Ч68
Binomial series      255Ч259
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      332Ч333
Cantor set      319Ч321 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 Ч 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 343Ч344
Chiliad      304
Chinese remainder theorem      373Ч375
Chudnovsky, D.V. and G.V.      237
Church, A.      52
Circle arc length of      232
Circle lattice points inside      221Ч223
Circle quadrature of      251Ч258 310 314 329
Circle rational points on      44Ч46 55
Cissoid of Diocles      328
Cohen, G.L.      101
coin tossing      225Ч230 248 254 263 283
Complex (Gaussian) integers      112
Composite numbers      7
Congruent numbers      369Ч377
Construction, straightedge and compass      142 310
Continued fractions      143Ч144 156 331
Contraction      180Ч181
Convex function      199
Convolution      164
Cox, D.A.      236
Coxeter, H.S.M.      24
Cube, duplication of      187 310 328
Cubes, sum of consecutive      78Ч81
Cummings, E.E.      1
Curve Bezier spline      284Ч286
Curve circle      44Ч46 55 221Ч223 232 251Ч258 310 314 329
Curve cissoid of Diodes      328
Curve devil's staircase      332Ч333
Curve ellipse      263 310
Curve folium of Descartes      56
Curve Gaussian      see УCurve normalФ
Curve hyperbola      196 245 289Ч290 310
Curve Koch      323Ч325 336
Curve lemniscate      231Ч232
Curve logarithmic spiral      148Ч153
Curve normal      213 327
Curve parabola      310Ч314
Curve snowflake      see УCurve KochФ
Curve spiral of Archimedes      148 311 328Ч329
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      276Ч281
Dense      44 46 158
density      291 295Ч300
Derangement      37 165
Descartes, R.      56 148 344
Devaney, R.L.      183
Devil's staircase      332Ч333
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      50Ч53
Diophantus of Alexandria      41 76
Diophantus of Alexandria Arithmetica      41 111
Diophantus of Alexandria, Book on Polygonal Numbers      76
Dirac delta function      276Ч281
Dirac, P.A.M.      276
Dirac, P.A.M., The Principles of Quantum Mechanics      276
Dirichlet's theorem      68 70 109Ч110
Dirichlet, P.G.Lejeune      104 245 306
Divergent series      223Ч225 355Ч356
Divine proportion      see УGolden ratioФ
Divisors, number of      244Ч245
Divisors, sum of      see also УPentagonal number theoremФ 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 204Ч205 209 213 239Ч240 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      231Ч232 263 310
Elliptic integral      231Ч232 263
Encke, J.F.      306
Equiangular spiral      see УLogarithmic spiralФ
Eratosthenes, sieve of      12 36 299
Erdos, P.      112 295
Error function      207
Euclid      312
Euclid Elements      27 43 101
Euclidean algorithm      26Ч29 35 128Ч130 144
Eudoxus      27
Eudoxus principle of      312 319
Euler $\phi$-function      36 117
Euler Ч 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 350Ч351 357
Exhaustion, method of      85 311
Expected value      163 225Ч230
Exponential function and series      195Ч197
Exponential generating function      161 167
Factorial      265Ч267
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 316Ч317
Fermat's last theorem      4 41 47 94
Fermat's little theorem      100Ч104 108 116Ч118
Fermat, P. de      see also УInfinite descentФ 55 75Ч76 114 330
Fibonacci (Leonardo of Pisa)      124
Fibonacci (Leonardo of Pisa) Liber abaci      124
Fibonacci numbers      35 71 123Ч135 138 154
Fibonacci search algorithm      138
Figurate numbers      74 342
Fixed point      172 180Ч181
Fixed Point Iteration      180
Foley, J.D.      285
Folium of Descartes      56
Formally undecidable propositions      51
Fractal      175 318 326Ч327
Freedman, D.      207
Frenicle de Bessy, B.      103
Fresnel integrals      273
Function $\pi$(x)      297
Function $\sigma$(n)      115 358Ч367
Function arithmetical      219
Function average value of      219Ч220 239
Function binomial      65Ч68
Function Cantor      332Ч333
Function convex      199Ч200
Function d(n)      244
Function dilogarithm      350
Function Dirac delta      276Ч281
Function elementary      266 268
Function error      207Ч213
Function Euler $\phi$      36 117
Function exponential generating      161 167
Function generating      159 163Ч165 167 169 259 261Ч263
Function logistic      181Ч184 7
Function peaking      274 278Ч280
Function r(n)      220Ч223 317
Function Riemann zeta      352Ч353
Functional equation      150Ч152 158 211
Fundamental Theorem of Arithmetic      see also УGaussian integersФ 23 48 371
Fundamental theorem of calculus      355
Fundamental theorem of information theory      133
Gage, P.      61
Gallon, F.      185 214Ч216
Gallon, F. Natural Inheritance      185 214
Gambler's ruin      162Ч164
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      112Ч113
Generating function      159 163Ч165 167 169 259 261Ч263
Geometric mean      20 186Ч195
Geometric series      310Ч324
Gilbert, W.S.      57
Gilbreath's conjecture      308
Girard, A.      111
Gleick, J.      322
Gnomon      81
Godel,K.      4 51
Goldbach's conjecture      5Ч6 58
Goldbach, C.      64 260
Golden ratio      140Ч153 159 171
Golden rectangle      144Ч148
Golden section      see УGolden ratioФ
Gosper, C.      238
Grabiner, J.V.      223
Graham, R.L.      12 295
Grandi, G.      223
Greatest common divisor      see also УEuclidean algorithmФ 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      287Ч288 295Ч296
1 2 3
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