Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.
Ce Livre est le quatrième du traité ; il est consacré aux bases de l analyse réelle. Il comprend les chapitres: 1. Dérivées; 2. Primitives et intégrales; 3. Fonctions élémentaires; 4. Équations différentielles; 5. Étude locale des fonctions; 6. Développements tayloriens généralisés. 7. Formule sommatoire d Euler-Maclaurin; 8. La function gamma.
Il contient également des notes historiques.
Ce volume est une réimpression de l édition de 1976.
Inhaltsverzeichnis
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.
