Analysis I: Convergence, Elementary functions

Cover......Page 1
Title......Page 3
Preface......Page 5
Contents......Page 19
I - Sets and Functions......Page 23
1 - Membership, equality, empty set......Page 29
2 - The set defined by a relation. Intersections and unions......Page 32
3 - Whole numbers. Infinite sets......Page 35
4 - Ordered pairs, Cartesian products, sets of subsets......Page 39
5 - Functions, maps, correspondences......Page 41
6 - Injections, surjections, bijections......Page 45
7 - Equipotent sets. Countable sets......Page 47
8 - The different types of infinity......Page 50
9 - Ordinals and cardinals......Page 53
$2. The logic of logicians......Page 61
0 - Introduction: what is a real number?......Page 67
1 - Algebraic operations and the order relation: axioms of R......Page 75
2 - Inequalities and intervals......Page 78
3 - Local or asymptotic properties......Page 81
4 - The concept of limit. Continuity and differentiability......Page 85
5 - Convergent sequences: definition and examples......Page 89
6 - The language of series......Page 98
7 - The marvels of the harmonic series......Page 103
8 - Algebraic operations on limits......Page 117
9 - Increasing sequences. Upper bound of a set of real numbers......Page 120
10 - The function log ?. Roots of a positive number......Page 125
11 - What is an integral?......Page 132
12 - Series with positive terms......Page 136
13 - Alternating series......Page 141
14 - Classical absolutely convergent series......Page 145
15 - Unconditional convergence: general case......Page 149
16 - Comparison relations. Criteria of Cauchy and d'Alembert......Page 154
17 - Infinite limits......Page 160
18 - Unconditional convergence: associativity......Page 161
19 - The Taylor series......Page 170
20 - The principle of analytic continuation......Page 180
21 - The function cot ? and the series  l/n2k......Page 184
22 - Multiplication of series. Composition of analytic functions. Formal series......Page 189
23 - The elliptic functions of Weierstrass......Page 200
1 - Limit values of a function. Open and closed sets......Page 209
2 - Continuous functions......Page 214
3 - Right and left limits of a monotone function......Page 219
4 - The intermediate value theorem......Page 222
5 - Limits of continuous functions......Page 227
6 - A slip up of Cauchy's......Page 233
7 - The uniform metric......Page 238
8 - Series of continuous functions. Normal convergence......Page 242
9 - Nested intervals, Bolzano-Weierstrass, compact sets......Page 247
10 - Cauchy's general convergence criterion......Page 250
11 - Cauchy's criterion for series: examples......Page 256
12 - Limits of limits......Page 261
13 - Passing to the limit in a series of functions......Page 263
14 - Derivatives of a function......Page 266
15 - Rules for calculating derivatives......Page 274
16 - The mean value theorem......Page 282
17 - Sequences and series of difFerentiable functions......Page 287
18 - Extensions to unconditional convergence......Page 292
19 - Partial derivatives and differentials......Page 295
20 - Differentiability of functions of class C1......Page 298
21 - Differentiation of composite functions......Page 301
22 - Limits of difFerentiable functions......Page 306
23 - Interchanging the order of differentiation......Page 309
24 - Implicit functions......Page 312
1 - Cartesian spaces and general metric spaces......Page 325
2 - Open and closed sets......Page 328
3 - Limits and Cauchy's criterion in a metric space; complete spaces......Page 330
4 - Continuous functions......Page 333
5 - Absolutely convergent series in a Banach space......Page 335
6 - Continuous linear maps......Page 338
7 - Compact spaces......Page 342
8 - Topological spaces......Page 344
1 - Rational exponents......Page 347
2 - Definition of real powers......Page 349
3 - The calculus of real exponents......Page 352
4 - Logarithms to base a. Power functions......Page 354
5 - Asymptotic behaviour......Page 355
6 - Characterisations of the exponential, power and logarithmic functions......Page 358
7 - Derivatives of the exponential functions: direct method......Page 361
8 - Derivatives of exponential functions, powers and logarithms......Page 364
9 - The number e. Napierian logarithms......Page 367
10 - Exponential and logarithmic series: direct method......Page 368
11 - Newton's binomial series......Page 373
12 - The power series for the logarithm......Page 381
13 - The exponential function as a limit......Page 390
14 - Imaginary exponentials and trigonometric functions......Page 394
15 - Euler's relation chez Euler......Page 405
16 - Hyperbolic functions......Page 410
17 - Absolutely convergent infinite products......Page 416
18 - The infinite product for the sine function......Page 419
19 - Expansion of an infinite product in series......Page 425
20 - Strange identities......Page 429
$4. The topology of the functions Arg(z) and Log(z)......Page 436
Index......Page 447