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Knopp K. — Theory and applications of infinite series
Knopp K. — Theory and applications of infinite series

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Название: Theory and applications of infinite series

Автор: Knopp K.

Аннотация:

Unusually clear and interesting classic covers real numbers and sequences, foundations of the theory of infinite series and development of the theory (series of valuable terms, Euler's summation formula, asymptotic expansions, other topics). Exercises throughout. Ideal for self-study.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1954

Количество страниц: 563

Добавлена в каталог: 06.12.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
"Almost all"      65
$\pi$      200 230
$\pi$, evaluation of      252—254
$\pi$, series for      214 215
$\varepsilon$-neighbourhood      20
$\zeta$-function, Remann's      346 444—446 491—492 531 538
Abel — Dini theorem      290
Abel's convergence test      314
Abel's limit theorem      177 349
Abel's partial summation      313 397
Abel's series      122 281 292
Abel's theorem, extension of      406
Abel, N.H.      122 127 211 281 290 299 313 314 321 424 459 467 556
Abscissa of convergence      441
Absolute convergence of products      222
Absolute convergence of series      136 seqq. 396
Absolute value      7 390
Adams, J.C.      183 256
Addition      5 30 32
Addition theorem for the binomial coefficients      209
Addition theorem for the exponential function      191
Addition theorem for the trigonometrical functions      199 415
Addition, term by term      48 70 134
Aggregate, closed      7
Aggregate, ordered      5
Alterations, finite number of, for sequences      47 70 95
Alterations, finite number of, for series      130 476
Alternating series      131 250 263 316 518
Ames, L.D.      244
Amplitude      390
Analytic functions      401 seqq.
Analytic functions, series of      429
Andersen, A.F.      488
Approach within an angle      404
Approximation      65 231
Archimedes      7 104
Area      169
Arithmetic, fundamental laws of      5
Arithmetic, means      72 460
Arrangement by squares, by diagonals      90
Arzela, S.      344
Associative law      5 6
Associative law for series      132
Asymptotic series (expansion, representation)      518 seq. 535
Asymptotically equal      68
Asymptotically proportional      68 247
Averaged comparison      464—466
Axiom, Cantor — Dedekind      26 33
Axioms of arithmetic      5
Bachmann, F.      2
Barnes, E.W.      546
Bernoulli's inequality      18
Bernoulli's numbers      183 203—204 237 479
Bernoulli's polynomials      523 534
Bernoulli, James and John      18 65 184 238 244 457 523 556
Bernoulli, Nicolaus      324
Bertrand, J.      282
Bieberbach, L.      478
Binary fraction      39
Binomial series      127 190 208—211 423—428
Binomial theorem      50 190
Bocher, M.      350
Bohr, H.      492
Bolzano — Weierstrass theorem      91 394
Bolzano, B.      87 91 394
Bonnet, O.      282
Boormann, J.M.      195
Borel, E.      320 471 477 543 549 561 556
Bound      16 158
Bound, upper, lower      96 159
Bounded functions      158
Bounded sequences      16 44 80
Breaking off decimals      249
Briggs, H.      58 257
Bromwich, l'A.      477 556
Brouncker, W.      104
Burkhardt, H.      353 375 546
Cahen, E.      290 441
Cajori, F.      322
Cantor — Dedekind axiom      26 33
Cantor, G.      1 26 33 68 355
Cantor, M.      12
Carmichael, R.D.      477
Catalan, E.      247
Cauchy — Toeplitz limit theorem      74 391
Cauchy's convergence theorem      120
Cauchy's double series theorem      143
Cauchy's inequality      408
Cauchy's limit theorem      72
Cauchy's product      147 179 488 512
Cauchy, A.L.      19 72 87 96 104 113 117 136 138 146—148 164 186 196 219 294 408 459 534 545 556
Centre of a power series      157
Cesaro, E.      292 318 322 466
Chapman, S.      477
Characteristic of a logarithm      58
Circle of convergence      402
Circular functions      59 see
Closed aggregate      7
Closed expressions for sums of series      232—240
Closed interval      20 162
Commutative law      5 6
Commutative law for products      227
Commutative law for series      138
Comparison tests of the first and second kinds      113 seq. 274
Completeness of the system of real numbers      34
complex numbers      see "Numbers"
Condensation test, Cauchy's      120 297
Conditionally convergent      139 226
Conditions F      464
Continued fractions      105
Continuity      161—162 171 174 404
Continuity of power series      174 177
Continuity of the straight line      26
Continuity, uniform      162
Convergence      64 78
Convergence for sequences      78—88
Convergence for series      110—120 124 282—290
Convergence for series of complex terms      396—401
Convergence for series of monotonely diminishing terms      120—124 294—296
Convergence for series of positive terms      116 117
Convergence for uniform convergences      332—338
Convergence of products      218 222
Convergence of series      101
Convergence, abscissa of      441
Convergence, absolute      136 seq. 222 396 435
Convergence, circle of      402
Convergence, conditional, unconditional      139 227
Convergence, criteria of      see "Convergence tests" "Main
Convergence, general remarks on theory of      298—305
Convergence, half-plane of      441
Convergence, interval of      153 327
Convergence, radius of      151 seqq.
Convergence, rapidity of      251 262 279 332
Convergence, region of      153
Convergence, systematization of theory of      305—311
Convergence, tests for Fourier series      361 364—372
Convergence, uniform      326 seq. 381 428
Convergent sequences      see "Sequences"
Cosine      199 seq. 384 414
Cotangent      202 seq. 417
Curves of approximation      329 330
D'Alembert, J.      458 459
De Morgan, A.      281
Decimal fractions      116 see
Decimal section      24 51
Dedekind's Test      315 348
Dedekind, R.      1 26 33 41
Dedekind, R., section      41
Dense      12
Diagonals, arrangement by      90
Difference      31 243
Difference-sequence      87
Differentiability      163
Differentiability of a power series      174—175
Differentiability, right hand, left hand      163
Differentiation      163—164
Differentiation, logarithmic      382
Differentiation, term by term      175 342
Dini's rule      367—368 371
Dini, U.      227 282 290 293 311 344
Dirichlet series      317 441
Dirichlet's integral      356 seq. 359
Dirichlet's rule      365 371
Dirichlet's test      315 347
Dirichlet, G. Lejeune-      138 329 347 356 375 547
Disjunctive criterion      118 308 309
Distributive law      6 135 146
Divergence      65 101 160 391
Divergence, definite      66 101 160 391
Divergence, indefinite      67 101 160
Divergence, proper      67
Divergent sequences      457 seqq.
Divergent series      457 seqq.
Division      6 32
Division of power series      180 seqq.
Division, term by term      48 71
Divisors, number of      446 451 517
Divisors, sum of      451 547
Doetsch, G.      478
Double series, analogue for products      437—438
Double series, theorem on      430
du Bois-Reymond's test      315 348
du Bois-Reymond, P.      68 87 96 301 304 305 353 355 379 556
Duhamel, J.M.C.      285
e      82 194—198
e, calculation of      251
Eisenstein, G.      180
Elliot, E.B.      314
Equality      28
Equivalence theorem of Knopp and Schnee      481
Ermakoff's test      296 seqq. 311
Error      65
Error, evaluation of      see "Evaluation of remainders"
Euclid      7 14 20 69
Eudoxus, postulate of      11 27 34
Eudoxus, theorem of      7
Euler's $\varphi$-function      451 548
Euler's constant      225 228 271 522 527 536 538 547 555
Euler's formulae      353 415 518 536
Euler's numbers      239
Euler's transformation of series      244—246 262—265 469 507
Euler, L.      1 82 104 182 193 204 211 228 238 243 244 262 353 375 384 385 413 415 439 445 457 468 507 518 535—536 556
Evaluation of $\pi$      252—254
Evaluation of e      251
Evaluation of logarithms      198 254—257
Evaluation of remainders      250 525 531—535
Evaluation of remainders, more accurate      259
Evaluation of roots      257—258
Evaluation of trigonometrical functions      258—259
Evaluation, numerical      247—260
Even functions      173
Everywhere convergent      153
Exhaustion, method of      69
Expansion of elementary functions in partial fractions      205—208 239 377 419
Expansion of infinite products      437
Expansion problem for asymptotic series      543 seqq.
Exponential function and series      148 191—198 411—414
Expressions for real numbers      230
Expressions for sums of series      230—273
Expressions for sums of series, closed      232—240
Extension      11 34
Faber, G.      546 556
Fabry, E.      267 556
Faculty series      446 seq.
Fatzius, N.      244
Fejer's integral      494
Fejer's theorem      493
Fejer, L.      493 496 546
Fibonacci's sequence      14 270 452
Finite number      15 16
Finite number of alterations      see "Alterations"
Fourier, J.B.      352 375
Fourier, J.B., coefficients, constants      354 361 362
Fourier, J.B., Riemann's theorem on      363
Fourier, J.B., series      350 seqq. 492
Frobenius, G.      184 490
Frullani      375
Fully monotone      263 264 305
Function      158 403
Function, interval of definition, limit, oscillation, upper and lower bounds of      158—159
Functions of a complex variable      403 seq.
Functions of a real variable      158 seq.
Functions, analytic      401 seq.
Functions, arbitrary      351—352
Functions, cyclometrical      213—215 421
Functions, elementary      189 seq.
Functions, elementary analytic      410 seq.
Functions, even, odd      173
Functions, integral      408 411
Functions, rational      189 seq. 410
Functions, regular      408
Functions, sequences of      326 seq. 429
Functions, trigonometrical      198 seq. 258 414
Fundamental law of integers      7
Fundamental law of natural numbers      6—7
Fundamental law of order      5 29
Fundamental laws of arithmetic      5 32
Gamma-function      225—226 385 440 530
Gaps in the system of rational numbers      3 seqq.
Gauss, K.F.      1 113 177 288 289 552 556
Geometric series      see "Series"
Gibbs' phenomenon      380 496
Glaisher, J.W.L.      180 555
Gmeiner, J.A.      399 556
Goldbach      458
Goniometry      415
Grandi, G.      133
Graphical representation      8 15 20 390
Gregory, J.      65 214
Gronwall, T.H.      380
Hadamard, J.      154 299 301 314
Hagen, J.      182
Hahn, H.      2 305
Half-plane of convergence      441
Hanstedt, B.      180
Hardy, G.H.      318 322 407 444 477 486 487 547 548
Harmonic series      see "Series"
Hausdorff, F.      478
Hermann, J.      131
History of infinite series      104
Hobson, E.W.      350
Holder, O.      465 490
Holmboe      459
Horn, J.      550
Hypergeometric series      289
Identically equal      16
Identity theorem for power series      172
Improper integral      see "Integral"
Induction, law of      6
Inequalities      7
Inequality of nests      29
Infinite number      15
Infinite series      see "Series"
Infinitely small      19
Innermost point      23 394
Integrability in Riemann's sense      166
integral      165 seq.
Integral test      294
Integral, improper      169—170
Integral, logarithmic      545
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