Author	Bourbaki, Nicolas. author
Title	Elements of Mathematics Functions of a Real Variable [electronic resource] : Elementary Theory / by Nicolas Bourbaki, Philip Spain
Imprint	Berlin, Heidelberg : Springer Berlin Heidelberg, 2004
Connect to	http://dx.doi.org/10.1007/978-3-642-59315-4
Descript	XIV, 338 p. online resource

This book is an English translation of the last French edition of Bourbaki's Fonctions d'une Variable Rรฉelle. 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 the study 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 -- ยง 1. First Derivative -- ยง 2. The Mean Value Theorem -- ยง 3. Derivatives of Higher Order -- ยง 4. Convex Functions of a Real Variable -- Exercises on ยง1 -- Exercises on ยง2 -- Exercises on ยง3 -- Exercises on ยง4 -- II Primitives and Integrals -- ยง 1. Primitives and Integrals -- ยง 2. Integrals Over Non-Compact Intervals -- ยง 3. Derivatives and Integrals of Functions Depending on a Parameter -- Exercises on ยง1 -- Exercises on ยง2 -- Exercises on ยง3 -- III Elementary Functions -- ยง 1. Derivatives of the Exponential and Circular Functions -- ยง 2. Expansions of the Exponential and Circular Functions, and of the Functions Associated with Them -- Exercises on ยง1 -- Exercises on ยง2 -- Historical Note (Chapters I-II-III) -- IV Differential Equations -- ยง 1. Existence Theorems -- ยง 2. Linear Differential Equations -- Exercises on ยง1 -- Exercises on ยง2 -- Historical Note -- V Local Study of Functions -- ยง 1. Comparison of Functions on a Filtered Set -- ยง 2. Asymptotic Expansions -- ยง 3. Asymptotic Expansions of Functions of a Real Variable -- ยง 4. Application to Series with Positive Terms -- Exercises on ยง1 -- Exercises on ยง3 -- Exercises on ยง4 -- Exercises on Appendix -- VI Generalized Taylor Expansions. Euler-Maclaurin Summation Formula -- ยง 1. Generalized Taylor Expansions -- ยง 2. Eulerian Expansions of the Trigonometric Functions and Bernoulli Numbers -- ยง 3. Bounds for the Remainder in the Euler-Maclaurin Summation Formula -- Exercises on ยง1 -- Exercises on ยง2 -- Exercises on ยง3 -- Historical Note (Chapters V and VI) -- VII The Gamma Function -- ยง 1. The Gamma Function in the Real Domain -- ยง 2. The Gamma Function in the Complex Domain -- Exercises on ยง1 -- Exercises on ยง2 -- Historical Note -- Index of Notation

1. Mathematics
2. Functions of real variables
3. Mathematics
4. Real Functions