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Hijab O. — Introduction to Calculus and Classical Analysis (Undergraduate Texts in Mathematics)
Hijab O. — Introduction to Calculus and Classical Analysis (Undergraduate Texts in Mathematics)

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Название: Introduction to Calculus and Classical Analysis (Undergraduate Texts in Mathematics)

Автор: Hijab O.

Аннотация:

This text is intended for an honors calculus course or for an introduction to analysis. Involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate majors. The book contains many remarkable features: * complete avoidance of /epsilon-/delta arguments by instead using sequences * definition of the integral as the area under the graph, while area is defined for EVERY subset of the plane * complete avoidance of complex numbers * heavy emphasis on computational problems * applications from many parts of analysis, e.g. convex conjugates, Cantor set, continued fractions, Bessel functions, the zeta functions, and many more * 344 problems with solutions in the back of the book


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1997

Количество страниц: 313

Добавлена в каталог: 05.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\approx$      230
$\beta$      207
$\gamma$      152
$\Gamma(x)$      163
$\mathbb{N}$      7
$\mathbb{Q}$      9
$\mathbb{R}$      3
$\mathbb{Z}$      9
$\pi$      95
$\psi(x)$      168 181 212 220
$\sigma(n)$      213
$\sim$      186 218
$\tau(x)$      206
$\theta$      211
$\theta_+$, $\theta_-$, $\theta_0$      212
$\zeta$      180
a | b      10
Absolute value      13
AGM      182
AGM curve      214
Algebraic order      58
angle      100
ArcCos      97
Archimedes’ constant      95
arcsin      97
arctan      99
Area      115
Area, additivity      123
Area, affine invariance      126
Area, dilation invariance      117
Area, monotonicity      118
Area, naive      116
Area, of a parallelogram      122
Area, of a rectangle      119
Area, of a triangle      122
Area, reflection invariance      118
Area, rotation invariance      122
Area, subadditivity      118
Area, translation invariance      116
Arithmetic progressions      223
Asymptotic equality      186 197 218
Asymptotic expansion      230 231
B(x, y)      193
Bailey — Borwein — Plouffe series, bell-shaped curve      189
Bernoulli function      206
Bernoulli numbers      206
Bernoulli series      206
Bessel function      171 181 196
beta function      193
Binomial coefficient      201
Binomial theorem      81
Binomial theorem, Newton’s generalization      91
Bound lower      4
Bound upper      4
Cantor set      111
Cantor set, area of the      126
Cauchy order      35
Cauchy — Schwarz inequality for integrals      152
Cauchy — Schwarz inequality in the plane      101
Chain rule      64
Codomain      2
Compact      39 41
Complement      2
Completeness property      4
Continued fractions      12 25 38 169 174
Continuity      45
Continuity, from the left      50
Continuity, from the right      50
Continuity, in two variables      178
Continuity, modulus of      48
Continuity, under the integral sign      178
Continuity, under the summation sign      180
Continuity, uniform      51
Continuity, uniform modulus of      51
Convex conjugate      81
Cosecant      99
cosh      86
Cosine      94
Cotangent      99
coth      205
Cover      115
Critical point      65
Critical value      65
de Morgan’s law      2
Decimal expansions      27
decimal point      28
Delta function      143
Derivative      61
Derivative, partial      189
Differentiation      61
Differentiation, under the integral sign      189
Differentiation, under the summation sign      195
DIGITS      27
Dilation      6 99 117
Dilation, centered      117
Discontinuity, jump      47
Discontinuity, mild      48
Discontinuity, removable      47
Discontinuity, wild      48
Distance      100
Domain      2
Dominated convergence theorem for integrals      170
Dominated convergence theorem for series      179
Domination      170
Duplication formula      201
Entire      89
entropy      201
Euclidean motion      99
Euler product      222
Euler — Maclaurin, derivative      227
Euler — maclaurin, formula      229
Euler’s constant      152
Even      9 67
Even, part      67
Expansion, base of      28
Expansion, binary      28
Expansion, decimal      27
Expansion, hexadecimal      28
Expansion, ternary      28
Factorial of a natural      28
Factorial of a real      164
Fatou’s Lemma      167
Fourier transform      181 196
Fractional part      10
Functional equation, AGM      186
Functional equation, theta      212
Functional equation, zeta      221
Functions      2
Functions, algebraic      55
Functions, bounded      127
Functions, concave      75
Functions, continuous      45
Functions, convex      75
Functions, decreasing      44
Functions, differentiable      61
Functions, elementary      229
Functions, exponential      57
Functions, increasing      44
Functions, logarithmic      57
Functions, monotone      44
Functions, negative part      128
Functions, nonnegative      127
Functions, positive part      128
Functions, power      55 56
Functions, rational      47
Functions, signed      128
Functions, smooth      74
Functions, strictly concave      75
Functions, strictly convex      75
Functions, strictly decreasing      44
Functions, strictly increasing      44
Functions, strictly monotone      44
Functions, theta      212
Functions, transcendental      56
Fundamental theorem      145
Gamma function      163
Gaussian, function      189
Gaussian, integral      188
Golden mean      38
Greatest integer      10
Heat equation      217
Homogeneity      182
Inductive      7
Inductive, hypothesis      8
Inductive, step      8
inf      4
Infimum      4
Infinite product      203
Infinite product for $\Gamma(x)$      210
Infinite product for sin      209
Infinite product for sinh      203
Integrability of nonnegative functions      128
Integrability of signed functions      129
Integral test      140
Integrals, additivity      134
Integrals, continuity      139
Integrals, continuity at the endpoints      137
Integrals, differentiability      144
Integrals, dilation invariance      132
Integrals, integration by parts      148
Integrals, linearity      147
Integrals, monotonicity      131
Integrals, of nonnegative functions      128
Integrals, of signed functions      129
Integrals, substitution      148
Integrals, translation invariance      132
Interior      113
Intermediate value property for continuous functions      50
Intermediate value property for derivatives      74
Intersection      2
interval      12
Interval, compact      13
Interval, open      13
Interval, punctured      42
Inverse function theorem for continuous functions      54
Inverse function theorem for differentiable functions      69
Irrationals      14
Laplace transform      172 195
Laplace transform of $\tau$      211
Laplace’s theorem      198
Limits from the left      43
Limits from the right      43
Limits lower      19
limits of functions      42
Limits of monotone sequences      18
Limits of sequences      19
Limits upper      19
Log-convex      167
L’Hopital’srule      73
Machin’s formula      108
Maclaurin series      85
Mappings      2
Mappings, bijective      2
Mappings, composition of      2
Mappings, injective      2
Mappings, invertible      3
Mappings, surjective      2
Maximum      5
Maximum principle      81
Maximum, global      65
Maximum, local      65
Mean value theorem      66
Mean value theorem, generalized      72
Mean, arithmetic      182
Mean, arithmetic-geometric      182
Mean, geometric      182
Method of exhaustion      110 153
Minimum      5
Minimum, global      65
Minimum, local      65
Monomial      46
Monotone convergence theorem for integrals      164
Monotone convergence theorem for series      166
Multiplicative      224
n!      28
nth root      14
Numbers, algebraic      58
Numbers, integers      9
Numbers, natural      7
Numbers, rational      9
Numbers, real      3
Numbers, transcendental      58
O(g(x))      230
Odd      9 67
Odd part      67
Parity      8
Partial sum      25
Partition      44
Partition, mesh of a      52
Paving      115
PI      95
Piecewise, constant      52 136
Piecewise, continuous      147
Piecewise, differentiable      153
Polar coordinates      98
Polynomial      46
Polynomial growth      199
Polynomial, coefficients of a      46
Polynomial, degree of a      46
Power series, differentiability of      89
PRIMES      10
Primitives      102
Primitives, integration by parts      104
Primitives, linearity      104
Primitives, substitution      105
Radius of convergence      88
RANGE      2
Ratio Test      88
Rectangle, compact      40 114
Rectangle, open      40 114
Reflection      6
Relation      2
Remainder      83
Remainder, Cauchy form      84
Remainder, integral form      150
Remainder, Lagrange form      84
Riemann integrable      142
Riemann sum      137 141
Root test      88
Rotation      99
Sawtooth formula      219
Secant      99
SEQUENCE      16
Sequences, bounded      19
Sequences, Cauchy      31
Sequences, convergent      23
Sequences, decreasing      17
Sequences, error      31
Sequences, increasing      17
Sequences, lower      18
Sequences, monotone      18
Sequences, of sets, increasing      154
Sequences, sub-      23
Sequences, upper      18
Series      25
Series, absolutely convergent      31
Series, alternating      30
Series, alternating version      37
Series, cauchy product      37
Series, comparison test      27
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