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Godement, Analysis I, Convergence, Elementary functions, Springer, 2004
Godement Analysis II《Analysis I: Convergence, Elementary Functions》
Preface
Ⅰ. Sets and Functions
Set Theory
Membership, equality, empty set
The set defined by a relation, intersections and unions
Whole numbers, infinite sets
Ordered pairs, cartesian products, sets of subsets
Functions, maps, correspondences
Injections, surjections, bijections
Equipotent sets, countable sets
The different types of infinity
Ordinals and cardinals
The Logic of Logicians
Ⅱ. Convergence: Discrete Variables
Convergent Sequences and Series
Introduction: what is a real number?
Algebraic operations and the order relation: axioms of R
Inequalities and intervals
Local or asymptotic properties
The concept of limit, continuity and differentiability
Convergent sequences: definition and examples
The language of series
The marvels of the harmonic series
Algebraic operations on limits
Absolutely Convergent Series
Increasing sequences, upper bound of a set of real numbers
The function log x, roots of a positive number
What is an integral?
Series with positive terms
Alternating series
Classical absolutely convergent series
Unconditional convergence: general case
Comparison relations, criteria of Cauchy and d'Alembert
Infinite limits
Unconditional convergence: associativity
First Concepts of Analytic Functions
The Taylor series
The principle of analytic continuation
The function cot x and the series ∑1/n^(2k)
Multiplication of series, composition of analytic functions, formal series
The elliptic functions of Weierstrass