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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:

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

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Предметный указатель

Inequality, Bessel      309
Inequality, Cauchy — Schwarz      14 177 294
Inequality, Minkowski      27 (Ex. 1.25)
Inequality, triangle      13 294
Infinite derivative      108
Infinite product      206
Infinite series      185
Infinite set      38
Infinitum      9
Infinity in C*      24
Infinity in R*      14
Inner Jordan content      396
Inner product      48 294
Integers      4
Integrable function, Lebesgue      260 407
Integrable function, Riemann      141 389
Integral equation      181
Integral test      191
Integral transform      326
Integration by parts      144 278
Integrator      142
Interior (or inner region) of a Jordan curve      447
Interior point      49 61
Interior, of a set      49 61
Intermediate-value theorem for continuous functions      85
Intermediate-value theorem for derivatives      112
Intersection of sets      41
Interval in $\mathbb R^n$       50 52
Interval in R      4
Inverse function      36
Inverse image      44 (Ex. 2.7) 81
Inverse-function theorem      372
Inversion formula for Fourier transforms      327
Inversion formula for Laplace transforms      342 (Ex. 11.38) 468
Irrational numbers      7
Isolated point      53
Isolated singularity      458
Isolated zero      452
Isometry      84
Iterated integral      167 287
Iterated limit      199
Iterated series      202
Jacobi, Carl Gustav Jacob (1804—1851)      351 368
Jacobian, determinant      368
Jacobian, matrix      351
Jordan arc      435
Jordan content      396
Jordan curve      435
Jordan curve theorem      447
Jordan theorem on Fourier series      319
Jordan, Camille (1838—1922)      312 319 396 435 447
Jordan-measurable set      396
Jump of a function      93
Jump, discontinuity      93
Kestelman, Hyman      165 182
Kronecker delta, $\delta_{ij}$       385 (Ex. 13.6)
Lagrange identity      27 (Ex. 1.23) 30 380
Lagrange multipliers      380
Lagrange, Joseph Louis (1736—1813)      27 30 380
Landau, Edmund (1877—1938)      31
Laplace transform      326 342 468
Laplace, Pierre Simon (1749—1827)      326 342 468
Laurent expansion      455
Laurent, Pierre Alphonse (1813—1854)      455
Least upper bound      9
Lebesgue bounded convergence theorem      273
Lebesgue criterion for Riemann integrability      171 391
Lebesgue dominated-convergence theorem      270
Lebesgue integral of complex functions      292
Lebesgue integral of real functions      260 407
Lebesgue measure      290 408
Lebesgue, Henri (1875—1941)      141 171 260 270 273 290 292 312 391 405
Legendre polynomials      336 (Ex. 11.7)
Legendre, Adrien-Marie (1752—1833)      336
Leibniz' formula      121 (Ex. 5.6)
Leibniz, Gottfried Wilhelm (1646—1716)      121
Length of a path      134
Levi monotone convergence theorem for sequences      267
Levi monotone convergence theorem for series      268
Levi monotone convergence theorem for step functions      265
Levi, Beppo (1875—1961)      265 267 268 407
Limit function      218
Limit in a metric space      71
Limit theorem of Abel      245
Limit, inferior      184
Limit, superior      184
Lindelof covering theorem      57
Lindelof, Ernst (1870—1946)      56
Line segment in $\mathbb R^n$       88
Linear function      345
Linear space      48
Linear space of functions      137 (Ex. 6.4)
Linearly dependent set of functions      122 (Ex. 5.9)
Liouville's theorem      451
Liouville, Joseph (1809—1882)      451
Lipschitz condition      121 (Ex. 5.1) 137 316
Lipschitz, Rudolph (1831—1904)      121 137 312 316
Littlewood, John Edensor      312
Local extremum      98 (Ex. 4.25)
Local property      79
Localization theorem      318
Logarithm      23
Lower bound      8
Lower integral      152
Lower limit      184
Mapping      35
Matrix      350
Matrix product      351
Maximum and minimum      83 375
Maximum-modulus principle      453 454
Mean convergence      232
Mean-value theorem for derivatives of real-valued functions      110
Mean-value theorem for derivatives of vector-valued functions      355
Mean-value theorem for integrals, multiple integrals      401
Mean-value theorem for integrals, Riemann integrals      160 165
Mean-value theorem for integrals, Riemann — Stieltjes integrals      160
Measurable function      279 407
Measurable set      290 408
Measure, of a set      290 408
Measure, zero      169 290 391 405
Mertens' theorem      204
Mertens, Franz (1840—1927)      204
Metric      60
Metric space      60
Minimum-modulus principle      454
Minkowski's inequality      27 (Ex. 1.25)
Minkowski, Hermann (1864—1909)      27
Modulus of a complex number      18
Moebius transformation      471
Moebius, Augustus Ferdinand (1790—1868)      471
Monotonic function      94
Monotonic sequence      185
Multiple integral      389 407
Multiplicative function      216 (Ex. 8.45)
n-measure      40S
Neighborhood      49
Neighborhood of infinity      15 25
Niven, Ivan M.      180 (Ex. 7.33)
Nonempty set      1
Nonmeasurable function      304 (Ex. 10.37)
Nonmeasurable set      304 (Ex. 10.36)
NonNegative      3
Norm of a function      102 (Ex. 4.66)
Norm of a partition      141
Norm of a vector      48
O, o, oh notation      192
One-to-one function      36
Onto      35
Open interval in $\mathbb R^n$       50
Open interval in R      4

Open mapping      370 454
Open mapping theorem      371 454
Open set in $\mathbb R^n$       49
Open set in a metric space      62
Open, covering      56 63
Operator      327
Order of pole      458
Order of zero      452
Order-preserving function      38
Ordered n-tuple      47
Ordered pair      33
Ordinate set      403 (Ex. 14.11)
Orientation of a circuit      447
Orthogonal system of functions      306
Orthonormal set of functions      306
Oscillation of a function      98 (Ex. 4.24) 170
Outer Jordan content      396
Parallelogram law      17
Parseval's formula      309 474
Parseval, Mark-Antoine (circa 1776—1836)      309 474
Partial derivative      115
Partial derivative of higher order      116
Partial sum      185
Partial summation formula      194
Partition of an interval      128 141
Path      88 133 435
Peano, Giuseppe (1858—1932)      224
Perfect set      67 (Ex. 3.25)
Periodic function      224 317
Pi, $\pi$ , irrationality of      180 (Ex. 7.33)
Piecewise-smooth path      435
Point in $\mathbb R^n$       47
Point in a metric space      60
Pointwise convergence      218
Poisson integral formula      473 (Ex. 16 5)
Poisson summation formula      332
Poisson, Simeon Denis (1781—1840)      332 473
Polar coordinates      20 418
Polygonal curve      89
Polygonally connected set      89
Polynomial      80
Polynomial, in two variables      462
Polynomial, zeros of      451 475
Power series      234
Powers of complex numbers      21 23
Prime number      5
Prime-number theorem      175 (Ex. 7.10)
Principal part      456
Projection      394
Quadratic form      378
Quadric surface      383
Quotient of complex numbers      16
Quotient of real numbers      2
Radius of convergence      234
Range of a function      34
Ratio Test      193
Rational function      81 462
Rational number      6
Real number      1
Real part      15
Rearrangement of series      196
Reciprocity law for Gauss sums      464
Rectifiable path      134
Reflexive relation      43 (Ex. 2.2)
Region      89
Relation      34
Removable discontinuity      93
Removable singularity      458
Residue      459
Residue theorem      460
Restriction of a function      35
Riemann condition      153
Riemann integral      142 389
Riemann localization theorem      318
Riemann sphere      17
Riemann theorem on singularities      475 (Ex. 16.22)
Riemann zeta function      192 209
Riemann — Lebesgue lemma      313
Riemann, Georg Friedrich Bernard (1826—1866)      17 142 153 192 209 312 313 318 389 475
Riesz — Fischer theorem      297 311
Riesz, Frigyes (1880—1956)      252 297 305 311
Righthand derivative      108
Righthand limit      93
Rolle's theorem      110
Rolle, Michel (1652—1719)      110
Root test      193
Roots of complex numbers      22
Rouche's theorem      475 (Ex. 16.14)
Rouche, Eugene (1832—1910)      475
Saddle point      377
scalar      48
Schmidt, Erhard (1876—1959)      335
Schoenberg, Isaac J.      224
Schwarz's lemma      474 (Ex. 16.13)
Schwarz, Hermann Amandus (1843—1921)      14 27 30 122 177 294
Schwarzian derivative      122 (Ex. 5.7)
Second mean-value theorem for Riemann integrals      165
Second-derivative test for extrema      378
Semimetric space      295
Separable metric space      68 (Ex. 3.33)
Sequence, definition of      37
Set algebra      40
Similar (equinumerous) sets      38
Simple curve      435
Simply connected region      443
Singularity      458
Singularity, essential      459
Singularity, pole      458
Singularity, removable      458
Slobbovian integral      249 (Ex. 9.17)
Space-filling curve      224
Spherical coordinates      419
Square-integrable functions      294
Stationary point      377
Stenographic projection      17
Step function      148 406
Stieltjes integral      140
Stieltjes, Thomas Jan (1856—1894)      140
Stone, Marshall H.      252
Strictly increasing function      94
Subsequence      38
Subset      1 32
Substitution theorem for power series      231
Sup norm      102 (Ex. 4.66)
Supremum      9
Symmetric quadratic form      378
Symmetric relation      43 (Ex. 2.2)
Tannery's theorem      299 (Ex. 10.7)
Tannery, Jules (1848—1910)      299
Tauber, Alfred      246
Tauberian theorem      246 251
Taylor's formula with remainder      113
Taylor's formula with remainder, for functions of several variables      361
Taylor's series      241 449
Taylor, Brook (1685—1731)      113 241 361 449
Telescoping series      186
Theta function      334
Tonelli — Hobson test      415
Tonelli, Leonida (1885—1946)      415
Topological property      84
Topological, mapping      84
Topology, point set      47
Total variation      129 178
Transformation      35 417
Transitive relation      43 (Ex. 2.2)
Triangle inequality      13 19 48 60 294
Trigonometric series      312
Two-valued function      86
Uncountable set      39
Uniform bound      221

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