Functions of a Real Variable

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.

From the reviews:



"Nicolas Bourbaki is the name given to a collaboration of mainly French mathematicians who wrote a series of textbooks that started from basics and aimed to present a complete picture of all essential mathematics. ... The Elements of Mathematics series is the result of this project. ... The translation is true to the original. ... should be part of any good library of mathematics books." (Partrick Quill, The Mathematical Gazette, March, 2005)

"The book under review is the latest installment in the translation into English of the voluminous Bourbaki exercise. ... Respectable mathematics libraries should have this book on their shelves. ... the book may be judged as an unqualified intellectual success and the publisher and the translator are to be congratulated on making it available in English." (Barry D.Hughes, The Australian Mathematical Society Gazette, Vol. 43 (1), 2005)