Apostol T. — Mathematical Analysis, Second Edition :: Электронная библиотека попечительского совета мехмата МГУ

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Apostol T. — Mathematical Analysis, Second Edition

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Название: Mathematical Analysis, Second Edition

Автор: Apostol T.

Аннотация:

It provides a transition from elementary calculus to advanced courses in real and complex function theory and introduces the reader to some of the abstract thinking that pervades modern analysis.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2

Год издания: 1974

Количество страниц: 492

Добавлена в каталог: 17.03.2014

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Предметный указатель

$(x_{1},...,x_{n})$ , point in $R^{n}$       47
$A \times B$ , cartesian product of A and B      33
$A(a; r_{1}, r_{2})$ , annulus with center a      438
$a_{n}=O(b_{n})$ , $a_{n}=o(b_{n})$ , big oh (little oh) notation      192
$B^{'}(a), B^{'}(a;r)$ , deleted neighborhood of a      457
$B_{M}(a;r)$ , ball in metric space M      61
$C^{*}$ , extended complex-number system      14
$det[a_{ij}]$ , determinant of matrix [ $a_{ij}$ ]      367
$D_{k}f$ , pertial derivative of f with respect to the kth coordinate      115
$D_{r,k}f$ , second-order partial derivative      116
$f \in C^{'}$ , the components of f have continuous first-order partials      371
$f \in C^{\infty}$ , f has derivatives of every order      241
$f \in R$ on [a,b], f is Riemann-integrable on [a,b]      142
$f \in R(\alpha)$ , on [a,b], f is Riemann-integrable with respect to $\alpha$ on [a,b]      141
, , right- (left-)hand limit of f at c      93
, convolution of f and g      328
$F: S \rightarrow T$ , function from S to T      35
$f^{'}(c)$ , derivative of f at c      104 114 117
$f^{'}(c;u)$ , directional derivative of f at c in the direction u      344
$f^{+}, f^{-}$ , positive (negative) part of a function f      261
$f^{-1}(Y)$ , inverse image of Y under f      44 81
$f_{n} \nearrow f$ a.e. on S, sequence { $f_{n}$ } increases on S and convergs to f a.e. on S      254
$F_{n}$ , sequence whose nth term is $F_{n}$       37
$J_{f}$ , Jacobian determinant of f      368
$l.i.m.\limits_{n\rightarrow \infty} f_{n} = f$ , {$f_{n}} converges in the mean to f      232
$L^{2}$ -norm      293 295
$L^{2}(I)$ , set of square-integrable functions on I      294
$n(\gamma, z)$ , winding number of a circuit $\gamma$ with respect to z      445
$Res\limits_{z=a} f(z)$ , residue of f at a      459
$R^{*}$ , extended real-number system      14
$R^{+}$ , $R^{-}$ , set of positive (negative) numbers      2
$R^{n}$ , n-dimensional Euclidean space      47
$S^{'}$ , set of accumulation points of S      54 62
$T_{c}$ , $f^{'}(c)$ , total derivative      347
$u_{k}$ , kth-unit coordinate vector      49
$V_{f}$ , total variation of f      129
$Z^{+}$ , set of positive integers      4
$\alpha \nearrow$ on [a,b], $\alpha$ is increasing on [a,b]      150
$\bar{S}$ , closure of S      54 62
$\bigcap$ , $\cap$ , intersection      41
$\bigcup$ , $\cup$ , union      40 41
$\chi_{S}$ , characteristic function of S      289
$\in$ , $\notin$ , belongs to (does not belong to)      1 32
$\int\limits_I f(x)dx$ , multiple integral      389 407
$\int\limits_\gamma f(x)dx$ , contour integral of f along $\gamma$       436
$\Lambda_{f}$ , lenght of a rectifiable path f      134
$\lim\limits_{x\rightarrow c^{+}}$ , $\lim\limits_{x\rightarrow c^{-}}$ , right- (left-)hand limit      93
$\mathcal P[a,b]$ , set of all partitions of [a,b]      128 141
$\mu(S)$ , Lebesgue measure of S      290
$\nabla f$ , gradient vector of f      348
$\Omega_{f}(T)$ , oscillation of f on a set T      98 170
$\omega_{f}(x)$ , oscillation of f at a point x      98 170
$\partial S$ , boundary of a set S      64
$\subseteq$ , is a subset of      1 33
$\underline{c}(S), \overline{c}(S)$ , inner (outer) Jordan content of S      396
$\|(f)\parallel$ , $L^{2}$ -norm of f      294 295
$\|x\parallel$ , norm or lenght of a vector      48
(a,+ $\infty$ ), [a, + $\infty$ ), (- $\infty$ , a), (- $\infty$ , a], infinite intervals      4
(a,b), [a,b], n-dimensional open (closed) interval      50 52
(a,b), [a,b], open (closed) interval with endpoints a and b      4
(f,g), inner product of functions f and g, in $L^{2}(I)$       294 295
(M,d), metric space M with metric d      60
a.e., almost everywhere      172
Abel, limit theorem      245
Abel, Neils Henrik, (1802-1829)      194 245 248
Abel, partial summation formula      194
Abel, test for convergence of series      194 248
Absolute convergence, of products      208
Absolute convergence, of series      189
Absolute value      13 18
Absolutely continuous function      139
Accumulation point      52 62
Additive function      45
Additivity of Lebesgue measure      291
Adherent point      52 62
Algebraic number      45
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      88
Area (content) of a plane region      396
Argand, Jean-Robert (1768-1822)      17
Argument of complex number      21
Arithmetic mean      205
Arzela, Cesare (1847-1912)      228 273
Arzela’s theorem      228 273
Associative law      2 16
Axioms for real numbers      1 2 9
B(a), B(a;r), open n-ball with center a, (radius r)      49
B-A, the set of points B but not in A      41
Ball, in $R^{n}$       49
Ball, in a metric space      61
Basis vectors      49
Bernoulli, James (1654-1705)      251 338 478
Bernoulli, numbers      251
Bernoulli, periodic functions      338
Bernoulli, polynomials      251 478
Bernstein's theorem      242
Bernstein, Sergei Natanovic (1880-1968)      242
Bessel function      475
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, Ossian (1319-1892)      165
Bonnet’s theorem      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 set      64
Boundary, point      64
Bounded, away from zero      130
Bounded, convergence      227 273
Bounded, function      83
Bounded, set      54 63
Bounded, variation      128
c(S), Jordan content of S      397
C, the set of complex numbers, the complex plane      16
Cantor intersection theorem      56
Cantor set      180
Cantor — Bendixon theorem      67
Cantor, Georg (1845-1918)      8 32 56 67 180 312
cardinal number      38
Carleson, Lennart      312
Cartesian product      33
Casorati — Weierstrass theorem      475
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      177 294
Cauchy — Schwarz inequality, for sums      14 27 30
Cauchy, Augustin-Louis (1789-1857)      14 73 118 177 183 207 222
Cauchy, inequalities      451
Cauchy, integral formula      443
Cauchy, integral theorem      439
Cauchy, principal value      277

Cauchy, product      204
Cauchy, Residue Theorem      460
Cauchy, 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
Closed, curve      435
Closed, interval      4 52
Closed, mapping      99
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
Completeness axiom      9
Complex number      15
Complex plane      17
Component of a metric space      87
Component of a vector      47
Component, interval      51
Composite function      37
Condensation point      67
Conditional convergent series      189
Conditional convergent series, rearrangement of      197
Conformal mapping      471
Conjugate complex number      28
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
Convolution integral      328
Convolution theorem, for Fourier transforms      329
Convolution theorem, for Laplace transforms      342
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
d(x,y), distance from x to y in metric space      60
Daniell, P.J. (1889-1946)      252
Darboux, Gaston (1842-1917)      152
de la Vallee-Poussin, C.J.(1866-1962)      312
de Moivre, Ham (1667-1754)      29
De Moivre’s Theorem      29
Decimals      11 12 27
Dedekind, Richard (1831-1916)      8
Deleted neighborhood      457
Dense set      68
Denumerable set      39
Derivative(s), directional      344
Derivative(s), of complex functions      117
Derivative(s), of real-valued functions      104
Derivative(s), of vector-valued functions      114
Derivative(s), partial      115
Derivative(s), total      347
Derived set      54 62
Determinant      367
Df(c), Jacobian matrix of f at c      351
Difference of two sets      41
Differentiation, of integrals      162 167
Differentiation, of sequences      229
Differentiation, of series      230
Dini's theorem, on Fourier series      319
Dini's theorem, on uniform convergence      248
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
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
e, irrationality of      7
Element of a set      32
Empty set      33
Equivalence, of paths      136
Equivalence, relation      43
Essential singularity      458
Euclidean, metric      48 61
Euclidean, space $\R^{n}$       47
Euclid’s lemma      5
Euler, Leonard (1707-1783)      149 192 209 365
Euler’s, constant      192
Euler’s, product for $\zeta(s)$       209
Euler’s, summation formula      149
Euler’s, theorem on homogeneous functions      365
Exponential form, of Fourier integral theorem      325

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