Overview

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.

Full Product Details

Author:   N. Bourbaki ,  P. Spain
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   2004 ed.
Dimensions:   Width: 15.50cm , Height: 2.00cm , Length: 23.50cm
Weight:   0.723kg
ISBN:  

9783540653400

ISBN 10:   3540653406
Pages:   338
Publication Date:   18 September 2003
Audience:   General/trade ,  General
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

I Derivatives.- § 1. First Derivative.- 1. Derivative of a vector function.- 2. Linearity of differentiation.- 3. Derivative of a product.- 4. Derivative of the inverse of a function.- 5. Derivative of a composite function.- 6. Derivative of an inverse function.- 7. Derivatives of real-valued functions.- § 2. The Mean Value Theorem.- 1. Rolle’s Theorem.- 2. The mean value theorem for real-valued functions.- 3. The mean value theorem for vector functions.- 4. Continuity of derivatives.- § 3. Derivatives of Higher Order.- 1. Derivatives of order n.- 2. Taylor’s formula.- § 4. Convex Functions of a Real Variable.- 1. Definition of a convex function.- 2. Families of convex functions.- 3. Continuity and differentiability of convex functions.- 4. Criteria for convexity.- Exercises on §1.- Exercises on §2.- Exercises on §3.- Exercises on §4.- II Primitives and Integrals.- § 1. Primitives and Integrals.- 1. Definition of primitives.- 2. Existence of primitives.- 3. Regulated functions.- 4. Integrals.- 5. Properties of integrals.- 6. Integral formula for the remainder in Taylor’s formula; primitives of higher order.- § 2. Integrals Over Non-Compact Intervals.- 1. Definition of an integral over a non-compact interval.- 2. Integrals of positive functions over a non-compact interval.- 3. Absolutely convergent integrals.- § 3. Derivatives and Integrals of Functions Depending on a Parameter.- 1. Integral of a limit of functions on a compact interval.- 2. Integral of a limit of functions on a non-compact interval.- 3. Normally convergent integrals.- 4. Derivative with respect to a parameter of an integral over a compact interval.- 5. Derivative with respect to a parameter of an integral over a non-compact interval.- 6. Change of order of integration.- Exercises on §1.- Exercises on §2.- Exercises on §3.- III Elementary Functions.- § 1. Derivatives of the Exponential and Circular Functions.- 1. Derivatives of the exponential functions; the number e.- 2. Derivative of logax.- 3. Derivatives of the circular functions; the number ?.- 4. Inverse circular functions.- 5. The complex exponential.- 6. Properties of the function ez.- 7. The complex logarithm.- 8. Primitives of rational functions.- 9. Complex circular functions; hyperbolic functions.- § 2. Expansions of the Exponential and Circular Functions, and of the Functions Associated with Them.- 1. Expansion of the real exponential.- 2. Expansions of the complex exponential, of cos x and sin x.- 3. The binomial expansion.- 4. Expansions of log(1 + x), of Arc tan x and of Arc sin x.- Exercises on §1.- Exercises on §2.- Historical Note (Chapters I-II-III).- IV Differential Equations.- § 1. Existence Theorems.- 1. The concept of a differential equation.- 2. Differential equations admitting solutions that are primitives of regulated functions.- 3. Existence of approximate solutions.- 4. Comparison of approximate solutions.- 5. Existence and uniqueness of solutions of Lipschitz and locally Lipschitz equations.- 6. Continuity of integrals as functions of a parameter.- 7. Dependence on initial conditions.- § 2. Linear Differential Equations.- 1. Existence of integrals of a linear differential equation.- 2. Linearity of the integrals of a linear differential equation.- 3. Integrating the inhomogeneous linear equation.- 4. Fundamental systems of integrals of a linear system of scalar differential equations.- 5. Adjoint equation.- 6. Linear differential equations with constant coefficients.- 7. Linear equations of order n.- 8 Linear equations of order n with constant coefficients.- 9 Systems of linear equations with constant coefficients.- Exercises on §1.- Exercises on §2.- Historical Note.- V Local Study of Functions.- § 1. Comparison of Functions on a Filtered Set.- 1. Comparison relations: I. Weak relations.- 2. Comparison relations: II. Strong relations.- 3. Change of variable.- 4. Comparison relations between strictly positive functions.- 5. Notation.- § 2. Asymptotic Expansions.- 1. Scales of comparison.- 2. Principal parts and asymptotic expansions.- 3. Sums and products of asymptotic expansions.- 4. Composition of asymptotic expansions.- 5. Asymptotic expansions with variable coefficients.- § 3. Asymptotic Expansions of Functions of a Real Variable.- 1. Integration of comparison relations: I. Weak relations.- 2. Application: logarithmic criteria for convergence of integrals.- 3. Integration of comparison relations: II. Strong relations.- 4. Differentiation of comparison relations.- 5. Principal part of a primitive.- 6. Asymptotic expansion of a primitive.- § 4. Application to Series with Positive Terms.- 1. Convergence criteria for series with positive terms.- 2. Asymptotic expansion of the partial sums of a series.- 3. Asymptotic expansion of the partial products of an infinite product.- 4. Application: convergence criteria of the second kind for series with positive terms.- 1. Hardy fields.- 2. Extension of a Hardy field.- 3. Comparison of functions in a Hardy field.- 4. (H)Functions.- 5. Exponentials and iterated logarithms.- 6. Inverse function of an (H) function.- Exercises on §1.- Exercises on §3.- Exercises on §4.- Exercises on Appendix.- VI Generalized Taylor Expansions. Euler-Maclaurin Summation Formula.- § 1. Generalized Taylor Expansions.- 1. Composition operators on an algebra of polynomials.- 2. Appell polynomials attached to a composition operator.- 3. Generating series for the Appell polynomials.- 4. Bernoulli polynomials.- 5. Composition operators on functions of a real variable.- 6. Indicatrix of a composition operator.- 7. The Euler-Maclaurin summation formula.- § 2. Eulerian Expansions of the Trigonometric Functions and Bernoulli Numbers.- 1. Eulerian expansion of cot z.- 2. Eulerian expansion of sin z.- 3. Application to the Bernoulli numbers.- § 3. Bounds for the Remainder in the Euler-Maclaurin Summation Formula.- 1. Bounds for the remainder in the Euler-Maclaurin summation formula.- 2. Application to asymptotic expansions.- 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.- 1. Definition of the Gamma function.- 2. Properties of the Gamma function.- 3. The Euler integrals.- § 2. The Gamma Function in the Complex Domain.- 1. Extending the Gamma function to C.- 2. The complements’ relation and the Legendre-Gauss multiplication formula.- 3. Stirling’s expansion.- Exercises on §1.- Exercises on §2.- Historical Note.- Index of Notation.

Reviews

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)

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