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preface
i - sets and functions
¡ì1. set theory
1 - membership, equality, empty set
2 - the set defined by a relation. intersections and unions
3 - whole numbers. infinite sets
4 - ordered pairs, cartesian products, sets of subsets
5 - functions, maps, correspondences
6 - injections, surjections, bijections
7 - equipotent sets. countable sets
8 - the different types of infinity
9 - ordinals and cardinals
¡ì2. the logic of logicians
ii - convergence: discrete variables
¡ì1. convergent sequences and series
0 - introduction: what is a real number?
1 - algebraic operations and the order relation: axioms of r
2 - inequalities and intervals
3 - local or asymptotic properties
4 - the concept of limit. continuity and differentiability
5 - convergent sequences: definition and examples
6 - the language of series
7 - the marvels of the harmonic series
8 - algebraic operations on limits
¡ì2. absolutely convergent series
9 - increasing sequences. upper bound of a set of real number
10 - the function log x. roots of a positive number
11 - what is an integral?
12 - series with positive terms
13 - alternating series
14 - classical absolutely convergent series
15 - unconditional convergence: general case
16 - comparison relations. criteria of cauchy and d'alembert
17 - infinite limits
18 - unconditional convergence: associativity
¡ì3. first concepts of analytic functions
19 - the taylor series
20 - the principle of analytic continuation
21 - the function cot x and the series ¡Æ 1/n2k
22 - multiplication of series. composition of analytic functions. formal series
23 - the elliptic functions of weierstrass
iii- convergence: continuous variables
¡ì1. the intermediate value theorem
1 - limit values of a function. open and closed sets
2 - continuous functions
3 - right and left limits of a monotone function
4 - the intermediate value theorem
¡ì2. uniform convergence
5 - limits of continuous functions
6 - a slip up of cauchy's
7 - the uniform metric
8 - series of continuous functions. normal convergence
¡ì3. bolzano-weierstrass and cauchy's criterion
9 - nested intervals, bolzano-weierstrass, compact sets
10 - cauchy's general convergence criterion
11 - cauchy's criterion for series: examples
12 - limits of limits
13 - passing to the limit in a series of functions
¡ì4. differentiable functions
14 - derivatives of a function
15 - rules for calculating derivatives
16 - the mean value theorem
17 - sequences and series of differentiable functions
18 - extensions to unconditional convergence
¡ì5. differentiable functions of several variables
19 - partial derivatives and differentials
20 - differentiability of functions of class c1
21 - differentiation of composite functions
22 - limits of differentiable functions
23 - interchanging the order of differentiation
24 - implicit functions
appendix to chapter iii
1 - cartesian spaces and general metric spaces
2 - open and closed sets
3 - limits and cauchy's criterion in a metric space; complete spaces
4 - continuous functions
5 - absolutely convergent series in a banach space
6 - continuous linear maps
7 - compact spaces
8 - topological spaces
iv - powers, exponentials, logarithms, trigonometric functions
¡ì1. direct construction
1 - rational exponents
2 - definition of real powers
3 - the calculus of real exponents
4 - logarithms to base a. power functions
5 - asymptotic behaviour
6 - characterisations of the exponential, power and logarithmic functions
7 - derivatives of the exponential functions: direct method
8 - derivatives of exponential functions, powers and logarithms
¡ì2. series expansions
9 - the number e. napierian logarithms
10 - exponential and logarithmic series: direct method
11 - newton's binomial series
12 - the power series for the logarithm
13 - the exponential function as a limit
14 - imaginary exponentials and trigonometric functions
15 - euler's relation chez euler
16 - hyperbolic functions
¡ì3. infinite products
17 - absolutely convergent infinite products
18 - the infinite product for the sine function
19 - expansion of an infinite product in series
20 - strange identities
¡ì4. the topology of the functions arg(z) and log z
index