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Apostol T.M. — Mathematical Analysis
Apostol T.M. — Mathematical Analysis

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Название: Mathematical Analysis

Автор: Apostol T.M.

Аннотация:

A glance at the table of contents will reveal that this textbook treats topics in analysis at the "Advanced Calculus" level. The aim has been to provide a development of the subject which is honest, rigorous, up to date, and, at the same time, not too pedantic. The book provides a transition from elementary calculus to advanced courses in real and complex function theory, and it introduces the reader to some of the abstract thinking that pervades modem analysis.
Thi edition differs from the first in many respects. Point set topology
is developed in the setting of general metric spaces as well as in Euclidean n-space,and two new chapters have been added on Lebesgue integration. The material on line integrals, vector analysis, and surface integraIs has been deleted. The order of some chapters has been rearranged, many sections have been completely rewritten,
and several new exercises have been added. (From preface)


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 5th edition

Год издания: 1981

Количество страниц: 492

Добавлена в каталог: 11.05.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$L^2$-norm      293 295
Abel, limit theorem      245
Abel, limit theorem, partial summation formula      194
Abel, limit theorem, test for convergence of series      194 248
Abel, Neils Henrik, (1802—1829)      194 245 248
Absolute convergence of products      208
Absolute convergence of series      139
Absolute value      13 18
Absolutely continuous function      139
Accumulation point      52 62
Additive function      45 (Ex. 2.22)
Additivity of Lebesgue measure      291
Adherent point      52 62
Algebraic number      45 (Ex. 2.15)
Almost everywhere      172 391
Analytic function      434
Annulus      438
Approximation theorem of Weierstrass      322
ARC      88 435
Arc length      134
Archimedean property of real numbers      10
Arcwise connected set      8S
Area (content) of a plane region      396
Argand, Jean-Robert (1768—1822)      17
Argument of complex number      21
Arithmetic mean      205
Arzela's theorem      228 273
Arzela, Cesare (1847—1912)      228 273
Associative law      2 16
Axioms for real numbers      1 2 9
Ball, in $\mathbb R^n$      49
Ball, in a metric space      61
Basis vectors      49
Bernoulli numbers      251 (Ex. 9.38)
Bernoulli periodic functions      338 (Ex. 11.18)
Bernoulli polynomials      251 (Ex. 9.38) 478
Bernoulli, James (1654—1705)      251 338 478
Bernstein's theorem      242
Bernstein, Sergei Natanovic      242
Bessel function      475 (Ex. 16.21)
Bessel inequality      309
Bessel, Friedrich Wilhelm (1784—1846)      309 475
beta function      331
Binary system      225
Binomial series      244
Bolzano — Weierstrass theorem      54
Bolzano's theorem      85
Bolzano, Bernard (1781—1848)      54 85
Bonnet's theorem      165
Bonnet, Emile (1819—1892)      165
Borel, Emile (1871—1938)      58
Bound, greatest lower      9
Bound, least upper      9
Bound, lower      8
Bound, uniform      221
Bound, upper      8
Boundary, of a point      64
Boundary, of a set      64
Bounded convergence      227 273
Bounded function      83
Bounded set      54 63
Bounded variation      128
Bounded, away from zero      130
Cantor intersection theorem      56
Cantor set      180 (Ex. 7.32)
Cantor — Bendixon theorem      67 (Ex. 3.25)
Cantor, Georg (1845—1918)      8 32 56 67 180 312
cardinal number      38
Carleson, Lennart      312
Cartesian product      33
Casorati — Weierstrass theorem      475 (Ex. 16.23)
Cauchy condition for products      207
Cauchy condition for sequences      73 183
Cauchy condition for series      186
Cauchy condition for uniform convergence      222 223
Cauchy — Riemann equations      118
Cauchy — Schwarz inequality, for inner products      294
Cauchy — Schwarz inequality, for integrals      177 (Ex. 7.16) 294
Cauchy — Schwarz inequality, for sums      14 27 30
Cauchy, Augustin-Louis (1789—1857)      14 73 118 177 183 207 222
Cauchy, inequalities      451
Cauchy, inequalities, integral formula      443
Cauchy, inequalities, integral theorem      439
Cauchy, inequalities, principal value      277
Cauchy, inequalities, product      204
Cauchy, inequalities, residue theorem      460
Cauchy, inequalities, sequence      73
Cesaro, Ernesto (1859—1906)      205 320
Cesaro, sum      205
Cesaro, summability of Fourier series      320
Chain rule, complex functions      117
Chain rule, matrix form of      353
Chain rule, real functions      107
Chain rule, vector-valued functions      114
Change of variables in a Lebesgue integral      262
Change of variables in a multiple Lebesgue integral      421
Change of variables in a Riemann integral      164
Change of variables in a Riemann — Stieltjes integral      144
Characteristic function      289
Circuit      435
Closed ball      67 (Ex. 3.31)
Closed curve      435
Closed interval      4 52
Closed mapping      99 (Ex. 4.32)
Closed region      90
Closed set      53 62
Closure of a set      53
Commutative law      2 16
Compact set      59 63
Comparison Test      190
Complement      41
Complete metric space      74
Complete orthonormal set      336 (Ex. 11.6)
Completeness axiom      9
Complex number      15
Complex plane      17
Component, interval      51
Component, interval, of a metric space      87
Component, interval, of a vector      47
Composite function      37
Condensation point      67 (Ex. 3.23)
Conditional convergent series      189
Conditional convergent series, rearrangement of      197
Conformal mapping      471
Conjugate complex number      28 (Ex. 1.29)
Connected metric space      86
Connected set      86
Content      396
Continuity      78
Continuity, uniform      90
Continuously differentiable function      371
Contour integral      436
Contraction, constant      92
Contraction, fixed-point theorem      92
Contraction, mapping      92
Convergence in a metric space      70
Convergence of a product      207
Convergence of a sequence      183
Convergence of a series      185
Convergence, absolute      189
Convergence, bounded      227
Convergence, conditional      189
Convergence, mean      232
Convergence, pointwise      218
Convergence, uniform      221
Converse of a relation      36
Convex set      66 (Ex. 3.14)
Convolution integral      328
Convolution theorem for Fourier transforms      329
Convolution theorem for Laplace transforms      342 (Ex. 11.36)
Coordinate transformation      417
Countable additivity      291
Countable set      39
Covering of a set      56
Covering theorem, Heine — Borel      58
Covering theorem, Lindeloef      57
Cramer's rule      367
Curve, closed      435
Curve, Jordan      435
Curve, piecewise-smooth      435
Curve, rectifiable      134
Daniell, P. J. (1889—1946)      252
Darboux, Gaston (1842—1917)      152
De Moivre's theorem      29 (Ex. 1.44)
De Moivre, Ham (1667—1754)      29
Decimals      11 12 27
Dedekind, Richard (1831—1916)      8
Deleted neighborhood      457
Dense set      68 (Ex. 3.32)
Denumerable set      39
Derivative(s) of real-valued functions      104
Derivative(s) of vector-valued functions      114
Derivative(s), directional      344
Derivative(s), of complex functions      117
Derivative(s), partial      115
Derivative(s), total      347
Derived set      54 62
Determinant      367
Difference of two sets      41
Differentiation of integrals      162 167
Differentiation of sequences      229
Differentiation of series      230
Dim's theorem, on Fourier series      319
Dim's theorem, on uniform convergence      248 (Ex. 9.9)
Dini, Ulisse (1845—1918)      248 312 319
Directional derivative      344
Dirichlet's test, for convergence of series      194
Dirichlet's test, for uniform convergence of series      230
Dirichlet, integrals      314
Dirichlet, kernel      317
Dirichlet, Peter Gustav Lejeune (1805—1859)      194 205 215 230 317 464
Dirichlet, product      205
Dirichlet, series      215 (Ex. 8.34)
disconnected set      86
Discontinuity      93
Discrete metric space      61
Disjoint sets      41
Disjoint sets, collection of      42
Disk      49
Disk of convergence      234
Distance function (metric)      60
Distributive law      2 16
Divergent product      207
Divergent sequence      183
Divergent series      185
Divisor      4
Divisor, greatest common      5
Domain (open region)      90
Domain of a function      34
Dominated Convergence Theorem      270
Dot product      48
Double sequence      199
Double series      200
Double, integral      390 407
Du Bois-Reymond, Paul (1831—1889)      312
Duplication formula for the Gamma function      341 (Ex. 11.31)
e, irrationality of      7
Element of a set      32
Empty set      33
Equivalence of paths      136
Equivalence relation      43 (Ex. 2.2)
Essential singularity      458
Euclid's lemma      5
Euclidean metric      48 61
Euclidean space $\mathbb R^n$      47
Euler's constant      192
Euler's product for $\zeta(s)$      209
Euler's summation formula      149
Euler's theorem on homogeneous functions      365 (Ex. 12.18)
Euler, Leonard (1707—1783}      149 192 209 365
Exponential form of Fourier integral theorem      325
Exponential form of Fourier series      323
Exponential function      7 19
Extended complex plane      25
Extended real-number system      14
Extension of a function      35
Exterior (or outer region) of a Jordan curve      447
Extremum problems      375
Fatou's lemma      299 (Ex. 10.8)
Fatou, Pierre (1878—1929)      299
Fejer's theorem      179 (Ex. 7.23) 320
Fejer, Leopold (1880—1959)      179 312 320
Fekete, Michel      178
Field of complex numbers      116
Field of real numbers      2
Finite set      38
Fischer, Ernst (1875—1954)      297 311
Fixed point of a function      92
Fixed-point theorem      92
Fourier coefficient      309
Fourier integral theorem      324
Fourier series      309
Fourier transform      326
Fourier, Joseph (1758—1830)      306 309 312 324 326
Fubini's theorem      410 413
Fubini, Guido (1879—1943)      405 410 413
Function, definition of      34
Fundamental theorem of algebra      15 451 475
Fundamental theorem of integral calculus      162
Gamma function, continuity of      282
Gamma function, definition of      277
Gamma function, derivative of      284 303
Gamma function, duplication formula for      341 (Ex. 11.31)
Gamma function, functional equation for      278
Gamma function, series for      304 (Ex. 10.31)
Gauss, Karl Friedrich (1777—1855)      17 464
Gaussian sum      464
Geometric series      190 195
Gibbs' phenomenon      338 (Ex. 11.19)
Global property      79
Goursat, Edouard (1858—1936)      434
Gradient      348
Gram — Schmidt process      335 (Ex. 11.3)
Gram, Jorgen Pedersen (1850—1916)      335
Greatest lower bound      9
Hadamard determinant theorem      386 (Ex. 13.16)
Hadamard, Jacques (1865—1963)      386
Half-open interval      4
Hardy, Godfrey Harold (1877—1947)      30 206 217 251 312
Harmonic series      186
Heine — Borel covering theorem      58
Heine's theorem      91
Heine, Eduard (1821—1881)      58 91 312
Hobson, Ernest William (1856—1933)      312 415
Homeomorphism      84
Homogeneous function      364 (Ex. 12.18)
Homotopic paths      440
Hyperplane      394
Identity theorem for analytic functions      452
Image      35
Imaginary part      15
Imaginary unit      18
Implicit-function theorem      374
Improper Riemann integral      276
Increasing function      94 150
Increasing sequence of functions      254
Increasing sequence of numbers      71 185
Independent set of functions      335 (Ex. 11.2)
Induction principle      4
Inductive set      4
1 2 3
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