Functions of a Real Variable Elementary Theory

This book is an English translation of the last French edition of Bourbaki’s Fonctions d'une Variable Réelle.

The first chapter is devoted to derivatives, Taylor expansions, the finite increments theorem, convex functions. In the second chapter, primitives and integrals (on arbitrary intervals) are studied, as well as their dependence with respect to parameters. Classical functions (exponential, logarithmic, circular and inverse circular) are investigated in the third chapter. The fourth chapter gives a thorough treatment of differential equations (existence and unicity properties of solutions, approximate solutions, dependence on parameters) and of systems of linear differential equations. The local study of functions (comparison relations, asymptotic expansions) is treated in chapter V, with an appendix on Hardy fields. The theory of generalized Taylor expansions and the Euler-MacLaurin formula are presented in the sixth chapter, and applied in the last one to thestudy of the Gamma function on the real line as well as on the complex plane.

Although the topics of the book are mainly of an advanced undergraduate level, they are presented in the generality needed for more advanced purposes: functions allowed to take values in topological vector spaces, asymptotic expansions are treated on a filtered set equipped with a comparison scale, theorems on the dependence on parameters of differential equations are directly applicable to the study of flows of vector fields on differential manifolds, etc.

I Derivatives.- § 1. First Derivative.- § 2. The Mean Value Theorem.- § 3. Derivatives of Higher Order.- § 4. Convex Functions of a Real Variable.- Exercises on §1.- Exercises on §2.- Exercises on §3.- Exercises on §4.- II Primitives and Integrals.- § 1. Primitives and Integrals.- § 2. Integrals Over Non-Compact Intervals.- § 3. Derivatives and Integrals of Functions Depending on a Parameter.- Exercises on §1.- Exercises on §2.- Exercises on §3.- III Elementary Functions.- § 1. Derivatives of the Exponential and Circular Functions.- § 2. Expansions of the Exponential and Circular Functions, and of the Functions Associated with Them.- Exercises on §1.- Exercises on §2.- Historical Note (Chapters I-II-III).- IV Differential Equations.- § 1. Existence Theorems.- § 2. Linear Differential Equations.- Exercises on §1.- Exercises on §2.- Historical Note.- V Local Study of Functions.- § 1. Comparison of Functions on a Filtered Set.- § 2. Asymptotic Expansions.- § 3. Asymptotic Expansions of Functions of a Real Variable.- § 4. Application to Series with Positive Terms.- Exercises on §1.- Exercises on §3.- Exercises on §4.- Exercises on Appendix.- VI Generalized Taylor Expansions. Euler-Maclaurin Summation Formula.- § 1. Generalized Taylor Expansions.- § 2. Eulerian Expansions of the Trigonometric Functions and Bernoulli Numbers.- § 3. Bounds for the Remainder in the Euler-Maclaurin Summation Formula.- Exercises on §1.- Exercises on §2.- Exercises on §3.- Historical Note (Chapters V and VI).- VII The Gamma Function.- § 1. The Gamma Function in the Real Domain.- § 2. The Gamma Function in the Complex Domain.- Exercises on §1.- Exercises on §2.- Historical Note.- Index of Notation.