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Ⅰ-Sets and Functions 1

1．Set Theory 7

1-Membership,equality,empty set 7

2-The set defined by a relation.Intersections and unions 10

3-Whole numbers.Infinite sets 13

4-Ordered pairs,Cartesian products,sets of subsets 17

5-Functions,maps,correspondences 19

6-Injections,surjections,bijections 23

7-Equipotent sets.Countable sets 25

8-The different types of infinity 28

9-Ordinals and cardinals 31

2．The logic of logicians 39

Ⅱ-Convergence:Discrete variables 45

1．Convergent sequences and series 45

0-Introduction: what is a real number? 45

1-Algebraic operations and the order relation:axioms of R 53

2-Inequalities and intervals 56

3-Local or asymptotic properties 59

4-The concept of limit.Continuity and differentiability 63

5-Convergent sequences:definition and examples 67

6-The language of series 76

7-The marvels of the harmonic series 81

8-Algebraic operations on limits 95

2．Absolutely convergent series 98

9-Increasing sequences. Upper bound of a set of real numbers 98

10-The function logx. Roots of a positive number 103

11-What is an integral? 110

12-Series with positive terms 114

13-Alternating series 119

14-Classical absolutely convergent series 123

15-Unconditional convergence:general case 127

16-Comparison relations.Criteria of Cauchy and d'Alembert 132

17-Infinite limits 138

18-Unconditional convergence:associativity 139

3．First concepts of analytic functions 148

19-The Taylor series 148

20-The principle of analytic continuation 158

21-The function cot x and the series Σ1/n2k 162

22-Multiplication of series.Composition of analytic func-tions.Formal series 167

23-The elliptic functions of Weierstrass 178

Ⅲ-Convergence:Continuous variables 187

1．The intermediate value theorem 187

1-Limit values of a function.Open and closed sets 187

2-Continuous functions 192

3-Right and left limits of a monotone function 197

4-The intermediate value theorem 200

2．Uniform convergence 205

5-Limits of continuous functions 205

6-A slip up of Cauchy's 211

7-The uniform metric 216

8-Series of continuous functions.Normal convergence 220

3．Bolzano-Weierstrass and Cauchy's criterion 225

9-Nested intervals,Bolzano-Weierstrass,compact sets 225

10-Cauchy's general convergence criterion 228

11-Cauchy's criterion for series:examples 234

12-Limits of limits 239

13-Passing to the limit in a series of functions 241

4．Differentiable functions 244

14-Derivatives of a function 244

15-Rules for calculating derivatives 252

16-The mean value theorem 260

17-Sequences and series of differentiable functions 265

18-Extensions to unconditional convergence 270

5．Differentiable functions of several variables 273

19-Partial derivatives and differentials 273

20-Differentiability of functions of class C1 276

21-Diferentiation of composite functions 279

22-Limits of differentiable functions 284

23-Interchanging the order of differentiation 287

24-Implicit functions 290

Appendix to Chapter Ⅲ 303

1-Cartesian spaces and general metric spaces 303

2-Open and closed sets 306

3-Limits and Cauchy's criterion in a metric space;complete spaces 308

4-Continuous functions 311

5-Absolutely convergent series in a Banach space 313

6-Continuous linear maps 316

7-Compact spaces 320

8-Topological spaces 322

Ⅳ-Powers,Exponentials,Logarithms,Trigonometric Functions 325

1．Direct construction 325

1-Rational exponents 325

2-Definition of real powers 327

3-The calculus of real exponents 330

4-Logarithms to base a.Power functions 332

5-Asymptotic behaviour 333

6-Characterisations of the exponential,power and logarithmic functions 336

7-Derivatives of the exponential functions:direct method 339

8-Derivatives of exponential functions,powers and logarithms 342

2．Series expansions 345

9-The number e.Napierian logarithms 345

10-Exponential and logarithmic series:direct method 346

11-Newton's binomial series 351

12-The power series for the logarithm 359

13-The exponential function as a limit 368

14-Imaginary exponentials and trigonometric functions 372

15-Euler's relation chez Euler 383

16-Hyperbolic functions 388

3．Infinite products 394

17-Absolutely convergent infinite products 394

18-The infinite product for the sine function 397

19-Expansion of an infinite product in series 403

20-Strange identities 407

4．The topology of the functions Arg(z)and Log z 414

Index 425