Analysis I Convergence, Elementary functions

I Sets and Functions.- §1. 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.- §2. The logic of logicians.- II - Convergence: Discrete variables.- §1. Convergent sequences and series.- 0 - Introduction: what is a real number?.- 1 - Algebraic operations and the order relation: axioms of ?.- 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.- §2. Absolutely convergent series.- 9 - Increasing sequences. Upper bound of a set of real numbers.- 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.- §3. First concepts of analytic functions.- 19 - The Taylor series.- 20 - The principle of analytic continuation.- 21 - The function cot x and the series ?1/n2k.- 22 - Multiplication of series. Composition of analytic functions Formal series.- 23 - The elliptic functions of Weierstrass.- III - Convergence: Continuous variables.- §1. 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.- §2. 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.- §3. 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.- §4. 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.- §5. 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.- §1. 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.- §2. 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.- §3. 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.- §4. The topology of the functions Arg(z) and Log z.

Roger Godement (October 1, 1921 - July 21, 2016) is known for his work in functional analysis, and also his expository books. He started as a student at the École normale supérieure in 1940, where he became a student of Henri Cartan. He started research into harmonic analysis on locally compact abelian groups, finding a number of major results; this work was in parallel but independent of similar investigations in the USSR and Japan. Work on the abstract theory of spherical functions published in 1952 proved very influential in subsequent work, particularly that of Harish-Chandra. The isolation of the concept of square-integrable representation is attributed to him. The Godement compactness criterion in the theory of arithmetic groups was a conjecture of his. He later worked with Jacquet on the zeta function of a simple algebra. He was an active member of the Bourbaki group in the early 1950s, and subsequently gave a number of significant Bourbaki seminars. He also took part in the Cartan seminar. He also wrote texts on Lie groups, abstract algebra and mathematical analysis.