Burn R.P. — Numbers and Functions: Steps to Analysis :: Электронная библиотека попечительского совета мехмата МГУ

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Burn R.P. — Numbers and Functions: Steps to Analysis

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Название: Numbers and Functions: Steps to Analysis

Автор: Burn R.P.

Аннотация:

The transition from studying calculus in high school to studying mathematical analysis in college is notoriously difficult. In this new edition of Numbers and Functions, Dr. Burn invites the student to tackle each of the key concepts, progressing from experience through a structured sequence of several hundred problems to concepts, definitions and proofs of classical real analysis. The problems, with all solutions supplied, draw readers into constructing definitions and theorems. This novel approach to rigorous analysis will enable students to grow in confidence and skill and thus overcome traditional difficulties in learning this subject.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd edition

Год издания: 2000

Количество страниц: 350

Добавлена в каталог: 11.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$\mathbb{Q}$ , countably infinite      4.25
$\mathbb{Q}$ , defined      2.1
Abel      5.91 5H 12.40 12H
Absolute convergence      5.66—5.67 2.77
Absolute value      2.52—2.64 3.33 3.54
Accumulation point (cluster point)      4.48
Alternating Series Test      5.62—5.63
Anti-derivative      10.52 10.54
Arc cosine      10.49—10.50
Arc length      10.43—10.48
Arc sine      8.44
Arc tangent      1.8 5.84 8.45 11.60—11.62
Archimedean order      3.19 3H
Archimedes      1H 3H 5H 10H
Area      10.1—10.9
Area under       10.6 10.9
Area under       10.1
Area under       10.2
Area under       10.3—10.5
Arithmetic mean      2.30—20.39
Ascoli      4H
Associative law      5.1
Asymptote      7.4
Barrow      10H
Berkeley, Bishop      8H
Bernoulli, Jacques      1H
Bernoulli, Jacques, inequality      2.29 3.21
Bernoulli, Johann      8H 9H
Bijection      6.5 6.6
Binomial theorem      1.5 5.98 5.112—5.113 8H 11H 12.45 12H
Blancmange function      12.46
Bolzano      2H 4.53 4H 5H 6H 7.12 7H 12H
Bolzano — Weierstrass theorem      4.46 4.53
Bound      3.48
Bound, lower: for sequence      3.5—3.7
Bound, lower: for set      4.53 7.8
Bound, upper: for sequence      3.5—3.7
Bound, upper: for set      4.53 7.8
Bounded sequence      3.5 3.13—3.14 3.62 4.43—4.47 4.83—4.84 5.25
Bounded sequence, convergent subsequence      3.80 4.46—4.47 4.83—4.84
Bounded sequence, eventually      3.13 3.62
Bounded sequence, monotonic      3.80 4.33—4.35
Bouquet and Briot      5H
Bourbaki      6H
Briggs      11H
Burkill      11.1 11.29
Cantor      4.53 4H 6H 9H App. App.
Cauchy      2H 3H 4H 5H 6.7 6.55 6H 7H 8.20 8H 9H 10.39 10.54 10.59 10H 11.32 11H 12.45 12H App. 3.25
Cauchy product      5.108—5.113 5H
Cauchy sequence      4.32 4.55—4.57 4H 5.18 7H
Cauchy — Hadamard formula      5.99—5.102
Cauchy's mean value theorem      9.25
Cauchy's ntr root test      5.35—5.39 5.80—5.83
Cavalieri      9H 10H
Chain rule      8.18 8.33 8H
Chinese Box Theorem      4.42
Circle of convergence      5.94 5.103—5.105 12.40
Circle, circumference of      2.39 12.48
Circular functions      10.39—10.62
Closed interval      3.78 7.8
Closed interval, nested      4.42
Cluster point      4.48 4.49—4.51 4.53 8.5
Comparison test, first      5.26
Comparison test, limit form of second      5.55
Comparison test, second      5.53
Completeness principle      4.31
Completing the square      2.40—4.42
Concave      9.24
Conditional convergence      5.71—5.76
Connected sets      7.8 7.21
Contiguous functions      6.96
Continued fraction      4.58
Continuity of composite functions      6.40
Continuity of contiguous functions      6.96
Continuity of differentiable functions      8.12
Continuity of mono tonic functions      7.7
Continuity of polynomials      6.29
Continuity of products of functions      6 26
Continuity of quotients of functions      6.54
Continuity of reciprocals of functions      6.52
Continuity of sums of functions      6.23
Continuity on closed intervals      7.29—7.35 7.43
Continuity on intervals      7.8—7.20 7.38
Continuity or limits of functions      12.20—12.23
Continuity, by limits      6.86 6.89
Continuity, by neighbourhoods, defined      6.64
Continuity, by sequences, defined      6.18
Continuity, by sequences, equivalent to neighbourhoods      6.67—6.68
Continuity, squeeze rule      6.36
Continuity, uniform      7.37—7.45
Continuous at one point      6.35 8.24
Continuous at one point, bounded in neighbourhood      6.69
Continuous at one point, positive in neighbourhood      6.70
Continuous everywhere, differentiable nowhere      12.46
Continuous function on closed intervals      7.29—7.35 7.43
Continuous function on intervals      7.8—7.10 7.37
Continuous function, absolute value of      6.41
Continuous function, invertible      7.22—7.27
Contraction mapping      9.18
Contradiction      2.12 4.9 4.18—4.21 5.93
Contrapositive      5.11 5.26 5.60 5.93
Convergence of absolutely convergent series      5.66—5.70 5.77
Convergence of bounded monotonic sequences      4.34—4.35
Convergence of Cauchy sequences      4.55—4.58
Convergence of geometric series      5.2 5.9 5.12 5.20
Convergence of infinite decimals      4.11—4.14 4.31
Convergence of sequences of functions: point wise      12.11—12.14
Convergence of sequences of functions: uniform      12.15—12.46
Convergence of sequences, definition      3.48 3.60
Convergence of series, definition      5.5
Convergence of Taylor scries      9.43 5.44
Convergence, conditional      5.71—5.75
Convergence, general principle of      4.57 5.18
Convergent sequences, absolute value rule      3.54
Convergent sequences, are bounded      3.62
Convergent sequences, difference rule      3.54
Convergent sequences, inequality rule      3.76
Convergent sequences, products of      5.54 5.55
Convergent sequences, quotients of      3.65—3.67
Convergent sequences, reciprocals of      3.65—3.66
Convergent sequences, scalar rule      3.54
Convergent sequences, shift rule      3.52
Convergent sequences, squeeze rule      3.54
Convergent sequences, subsequence rule      3.54
Convergent sequences, sums of      3.54 3.55
CONVERSE      2.4
Convex functions      9.24
Cosine, definition      9.40 11.50
D'Alembert      3H 5H 8H
d'Alembert, test for sequences      3.40—3.41 3.70—3.73 3H 11.35
d'Alembert, test for series      3H 5.40—5.50 5.68—5.69 5.79 5.95 5.97 5H
Darboux      7H 9.12 10.20 10.50—10.51 10H
de l'Hopital      8H 9.26 9H 11.32
de la Chapelle      3H
De Morgan      1H
De Sarasa      10H 11H
Decimals      3.51 4.11—4.17 4.22 4.31
Decimals, infinite      3.51 4.14 4.23 4.30
Decimals, non-recurring      4.17
Decimals, recurring      4.11—4.16
Decimals, terminating      3.51 4.9 4.11 4.22 4.31
Decreasing function      5.56—4.57
Decreasing sequence      3.4 3.6
Dedekind      4H App.
Dense sets      4.4—4.11
Derivative      8.5
Derivative, inverse function      8.39—8.42
Derivative, second      8.35—8.37 9.22 9.29
Derived function      8.26—8.28
Descartes      8H

Differentiable function are continuous      8.12
Differentiable function at one point      8.24
Differentiable function, chain rule for      8.18
Differentiable function, products of      8.13
Differentiable function, quotients of      8.17
Differentiable function, sums of      8.10
Dini      9H
Dirichlet      3H 5H 6.20 6H 7H 10.11 10H 12.27
Discontinuity      6.16 6.18—6.20 12.3—12.35
Discontinuity, jump      6.18 6.79—6.80
Discontinuity, removable      6.79 6.82
Divergent series      5.11 5.28—5.30 5.38 5.47 5.75
Domain of function      6.1 6.3
du Bois — Reymond      4.34 4.57 4H
e      2.43—2.49 2H 4.36 5.23 11.27—11.28 11.32
Equivalent propositions      4.32 4.46 4.56 4.79
Euclid      2.32 2H 3H 4H
Euler      2H 5.27 5.56 5H 6H 8H 11.32 11H
Euler's constant      5.56
Eventually, in sequence      3.13 3.17 3.18 3.28
Exponential function      5.110 9.38 11.28 11.32—11.33 11.37
Extending of continuous function on $\mathbb{Q}$       7.46—7.48
Fermat      1H 8H 10.1 10.3 10H
Field      App. 1
Field, Archimedean ordered      App. 1
Field, complete Archimedean ordered      App. 1
Field, ordered      2.1—2.29 App.
First comparison test for series      5.26
Fixed point      7.20 9.18
Floor term      3.15—3.16
FOURIER      5H 6H 10H
Function      6.1 6H
Function, arrow diagram of      6.2
Function, bounded      7.31—7.32
Function, composite      6.38 6.40
Function, constant      6.21 8.6 9.17
Function, continuous      6.18
Function, continuous, bounded      6.69 7.31—7.32 7.34
Function, continuous, positive      6.70
Function, differentiable      8.4
Function, Dirichlet's      6.20
Function, domain of      6.1
Function, fixed points      7.20 9.18
Function, identity      6.22
Function, integer      3.19 6.12
Function, integrable      5.57
Function, inverse      6. 7.22—7.27
Function, is continuous      8.12
Function, monotonic      5.57 7.1—7.7 7.22—7.27
Function, one to one      6.3
Function, onto      6.2
Function, point wise limits      6.9—6.11 12.5
Function, range of      6.1 6.2 6.3
Function, real      6.1
Function, ruler      6.72 10.36
Function, step      10.12—10.15
Function, sum of      6.23 8.10
Function, uniform limits      12.15—12.46
Function, waterfall      App. 3.22
Fundamental Theorem of Arithmetic      4.1 3.18
Fundamental theorem of calculus      10.54 10H
Galileo      4H 10H
Gauss      2.38 4H 5H
Geometric mean      2.30—2.39
Geometric progression      3.21 2.39 2.42 5.2 5.5—5.9 5.12 5.32 5.40 5.86
Grassmann      1H
Greatest lower bound      4.71—4.78 4.81
Gregory of St Vincent      10.6 10.9 10H 11H
Gregory, James      5H 9H
hadamard      5H
Hardy      3H 8H 11.1 11.39
Harmonic series      5.30 5H
Harnack      2H 4H 11H
Heine      4.53 4H 6H 7H
Helmholtz      1H
Heron      2.37
Herschel      6H
hilbert      3H
Hopilal      see “de l'Hopital”
HORNER      4.37
Improper integrals      10.61—10.68
Increasing function      7.1
Increasing sequence      3.4
Indefinite integrals      10.49—10.51
Indices, laws of      11.1 11.6 11.8 11.20
Induction, Principle of Mathematica      1.1—1.8
Infimum      4.71—4.78 4.81—4.84 7.5—7.7
Infinite decimal      3.51 4.11—4.17 4.31
infinity      4.22—4.30
Infinity, countable      4.24—4.27
Infinity, function tends to      6.100—6.101
Infinity, sequence tends to      3.17 3.20—3.23
Infinity, uncountable      4.23 4.28—4.30
Injection      6.3
INTEGER function      3.19 3.51 4.6
Integrable functions, continuous functions      10.39
Integrable functions, modulus of      10.34—10.35
Integrable functions, monotonic functions      10.7—10.8
Integrable functions, scalar multiple of      10.26
Integrable functions, step functions      10.12—10.15
Integrable functions, sum of two      10.10 10.29
Integral test for convergence      5.57 5H
Integral, improper      10.61—10.68
Integral, indefinite      10.50—10.51
Integral, lower      10.16—10.17 10.20—10.22
Integral, of continuous function      10.43 10.45
Integral, of step function      10.12—10.15
Integral, Riemann      10.23—10.24
Integral, upper      10.18—10.22
Integration by parts      10.57—10.59
Integration by substitution      10.60
Intermediate Value Theorem      7.12—7.21 App.
Intermediate Value Theorem for derivatives      9.12
Intervals      4.8 7.8—7.10 7.21
Intervals, closed      3.78 7.9 7.12—7.20 7.29—7.34
Intervals, end points of      6.91—6.92 8.46
Intervals, nested      4.42
Intervals, open      3.78 7.8
Inverse functions      6.6
Inverse of continuous functions      7.22—7.27
Irrational numbers      4.18—4.21 5.24
Iterative formula      9.18
JORDAN      2H
Jump discontinuity      6.18 6.79 6.80 10.56
Klein      11H App.
Koch      3.82
Koerner      8.42
L'Hopital      see “de l'Hopital”
l'Huilier      3H 6H
Lacroix      3H
Lagrange      8H 9H
Laurent scries      App. 1 App.
Least upper bound      4.59—4.66 4.80—4.82
Leathern      3H 6H
Leibniz      5H 8H 10.43 10.57 10.60 11H
Less than      2.1
Lim sup and lim inf      4.83—4.84 5.102
Limit of composite function      6.99
Limit of function as $x \rightarrow \infty$       6.100
Limit of sequence      3.48 3.60
Limit of, modulus of      6.93
Limit of, one-sided      6.73 6.76 7.5—7.6
Limit of, products of      6.93
Limit of, reciprocal of      6.93
Limit of, sequence definition equivalent to neighbourhood definition      6.84—6.85 6.87—6.88
Limit of, sums of      6.93
Limit of, two-sided      6.86
Liouvill      4H
Lipschitz condition      7.42 9.18 10.50 11.18
Littlewood      App. 2

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