← Retour

Mathematical analysis 1

par: Bekkai, Sanaa , Sofiane Messirdi & Abdellah Gherbi
Disponible
Pages 432
Dimensions 16*23,5
Publication 2026-10-15
Éditeur —
Papier : 1 950.00 DA Numérique: 1 950.00 DA

Mathematical analysis is a universal language that enables us to describe, under- stand, and model a wide variety of phenomena in science and engineering. It is therefore a fundamental component of scientific university programs, where it pro- vides the theoretical foundations for the study of more advanced concepts in sub- sequent years. For this reason, students should become familiar with its language, methods, and fundamental principles from the very beginning of their university studies. Although mathematical analysis is a central and far-reaching branch of mathematics, its basic concepts can be understood and mastered progressively through a clear presentation, appropriate examples, and regular practice. This book is intended primarily for first-year students enrolled in Scientific Univer- sity Foundation Programs within the LMD (Licence–Master–Doctorate) system, as well as for second-year students in applied sciences. It is also designed to serve as a reference for students pursuing engineering studies and may prove useful to those studying economics, agricultural sciences, medicine, and biology. The book covers the first part of a mathematical analysis course taught to first-year students in a variety of programs, including exact sciences, mathematics and computer science, science and technology, materials science, natural and life sciences, and engineering sciences. The structure and content of this book have been shaped by our teaching experience and by the questions, observations, and difficulties that naturally arise in the class- room. Our aim is to provide students and instructors with a textbook that reflects the progression of a university-level course while incorporating the explanations, com- ments, and clarifications that are often developed during lectures. We have sought to combine a systematic presentation of mathematical concepts with a pedagogical approach that facilitates understanding and encourages students to develop sound mathematical reasoning. The topics covered in this volume follow the official syllabus of the Analysis I teach- ing unit, as established by the Ministry of Higher Education and Scientific Research (MESRS), while taking into account the broader context of university-level scientific education. Organized into eight chapters, this first volume focuses primarily on the theory of real-valued functions of a real variable. It covers the algebraic and order structures of the real numbers, the field of complex numbers, the convergence of numerical se- quences, limits of functions, continuity, differentiability, and Taylor expansions near a point and at infinity. These topics are developed progressively, allowing students to build a coherent understanding of the fundamental principles of mathematical analysis. Note. The fundamental concepts of set theory and mathematical logic are not covered in this book, as they are generally introduced in algebra courses. Readers are therefore expected to be familiar with these basic notions. Throughout the book, definitions and fundamental concepts are introduced rigor- ously and illustrated by carefully chosen examples. We give particular importance to simple and classical situations, which provide a natural starting point for devel- oping intuition and understanding more abstract ideas. The principal properties are then formulated as theorems, lemmas, and propositions, with proofs presented in a clear and accessible manner. Our approach seeks to maintain mathematical rigor while helping students understand not only the statements of results but also the reasoning underlying them. Each chapter provides a comprehensive yet concise presentation of the material, followed by 30 exercises. The book therefore contains a total of 240 exercises, all accompanied by detailed solutions. These exercises are designed to reinforce the fundamental concepts, develop mathematical reasoning, and help students acquire the skills required to solve problems independently. Most are accessible to stu- dents who have carefully studied the corresponding material and generally do not rely on sophisticated tricks or exceptional ingenuity. Our intention is to encourage steady progress and enable students to solve a substantial proportion of the exercises through a sound understanding of the underlying principles. Alongside these introductory exercises, we have included more demanding prob- lems intended to deepen students’ knowledge and familiarize them with essential techniques of real analysis. The exercises and their solutions complement the the- oretical exposition, providing opportunities to apply definitions, use established re- sults, and develop a more precise and rigorous mathematical approach. We hope that this book will provide students with a solid foundation for further studies in mathematics and related scientific disciplines. We also hope that it will serve as a practical teaching resource for instructors and contribute to making the study of mathematical analysis a rewarding and intellectually stimulating experi- ence. We warmly recommend it to students beginning their university studies in analysis and to instructors involved in teaching introductory courses in this funda- mental subject. Bekkai Messirdi1 bmessirdi@yahoo.fr Sofiane Messirdi2 messirdi.sofiane@hotmail.fr Abdellah Gherbi3 gherbi_abdellah@esgee-oran.dz Sanaa Messirdi4 blahasanaa@yahoo.fr

Afficher la table des matières
Introduction
1 The Field of Real Numbers
1.1 Generalities
1.1.1 Order Relation on a Set
1.1.2 Upper and Lower Bounds
1.2 The Field of Real Numbers ℝ
1.2.1 Properties of Addition in ℝ
1.2.2 Properties of Multiplication in ℝ
1.2.3 ℝ is a Totally Ordered Field
1.2.4 Some Useful Formulas
1.2.5 Intervals of ℝ
1.2.6 Upper Bounds, Lower Bounds, Supremum and Infimum in ℝ
1.2.7 Axioms of the Supremum and Infimum
1.2.8 Characterization of the Supremum and Infimum
1.2.9 The Absolute Value of a Real Number
1.2.10 Integer Part and n-th Root of a Real Number
1.2.11 Decimal Expansions
1.2.12 Topology of the Real Line
1.3 Exercises with Solutions
2 The Field of Complex Numbers
2.1 The Concept of a Field
2.2 Construction of the Field of Complex Numbers
2.2.1 Notation and conventions
2.2.2 Elementary Properties of Complex Numbers
2.3 Powers and Roots
2.3.1 Complex Exponential
2.3.2 nth Roots of a Complex Number
2.3.3 Solving Quadratic Equations in ℂ
2.4 Trigonometric Applications
2.4.1 Review of Trigonometry
2.4.2 Addition Formulas.
2.4.3 Double-Angle Formulas.
2.4.4 Expansion of cos na, sin na and tan na
2.4.5 Linearization of cos^p a and sin^p a
2.4.6 Solving the Equation a cos x + b sin x = c
2.5 Complex Numbers and Transformations
2.5.1 Complex Expression of a Translation
2.5.2 Complex Expression of a Homothety
2.5.3 Complex Expression of a Rotation
2.5.4 Similarities in the Complex Plane
2.6 Exercises with Solutions
3 Sequences of Real Numbers
3.1 Definitions and Generalities
3.2 Convergence of a Numerical Sequence
3.3 Subsequences
3.4 Bounded Sequences
3.5 Bounds and Algebraic Operations on Sequences
3.6 Extension to Infinite Limits
3.7 Some Examples of Sequences
3.7.1 The Sequence n^r, r ∈ ℚ
3.7.2 The Arithmetic Sequence
3.7.3 The Sequence a^n, a ∈ ℝ
3.7.4 The Sequence a^n/n^p, a ∈ ℝ, a > 1, p ∈ ℕ
3.7.5 The Sequence a^(1/n), a ∈ ℝ, a > 0
3.7.6 The Sequence n^(1/n), n ∈ ℕ*
3.7.7 The Sequence (a^n/n!)_{n∈ℕ*}, a ∈ ℝ
3.8 Monotone Sequences
3.9 Adjacent Sequences
3.10 Recursive Sequences
3.10.1 Possible Limits of a Recursive Sequence
3.10.2 Monotonicity of a Recursive Sequence
3.11 Cauchy Sequences
3.12 Bolzano–Weierstrass Theorem
3.13 Exercises with Solutions
4 Real-Valued Functions of a Real Variable
4.1 General Properties of Numerical Functions
4.1.1 Domain, Image, Preimage, and Graph
4.1.2 Operations on ℝ^X
4.1.3 Symmetry, Periodicity, and Monotonicity
4.2 Limits of Functions
4.2.1 Right-Hand and Left-Hand Limits
4.2.2 Properties of Limits
4.2.3 Operations on Limits
4.3 Infinite Limits
4.4 Some Standard Limits
4.5 Some Important Results
4.6 Some Standard Functions
4.6.1 Natural Logarithmic Function
4.6.2 Logarithmic Function to Base a
4.6.3 Exponential Function
4.6.4 Power Functions
4.7 Exercises with Solutions
5 Continuity of Real-Valued Functions
5.1 Continuity of Real-Valued Functions
5.1.1 Properties of Continuous Functions
5.1.2 Fundamental Theorems on Continuous Functions
5.1.3 Strictly Monotone Continuous Functions
5.1.4 Fixed-Point Theorem
5.2 Exercises with Solutions
6 Usual Inverse Functions
6.1 n-th Root Functions
6.2 Inverse Trigonometric Functions
6.2.1 The Function arcsin x
6.2.2 The Function arccos x
6.2.3 The Function arctan x
6.2.4 The Function arccot x
6.2.5 Examples
6.3 Hyperbolic Functions
6.3.1 Standard Limits
6.4 Inverse Hyperbolic Functions
6.4.1 The Function arg sinh x
6.4.2 The Function arg cosh x
6.4.3 The Function arg th x
6.5 Exercises with Solutions
7 Differentiable Functions
7.1 Derivative at a Point
7.2 Right and Left Derivatives at a Point
7.3 Differentiability and Continuity
7.4 Operations on Differentiable Functions
7.5 Summary Table of Derivatives of Common Functions
7.6 Higher-Order Derivatives
7.7 Properties of Differentiable Functions
7.8 Rolle's Theorem
7.9 Mean Value Theorem
7.9.1 An Alternative Form of the Mean Value Formula
7.9.2 General Form of the Mean Value Formula
7.10 Monotonicity of a Differentiable Function
7.11 L'Hospital's Rule
7.12 Taylor Formulas and Their Variants
7.12.1 Other Forms of Taylor's Formula of Order (n + 1)
7.12.2 Taylor (and Maclaurin) Formulas for Some Common Functions
7.13 Taylor–Lagrange Inequality
7.14 Exercises with Solutions
8 Taylor Expansions
8.1 Comparison of Functions Near a Point
8.2 Taylor Expansions
8.3 Some Taylor Expansions Using Maclaurin's Formula
8.4 Operations on Taylor Expansions
8.4.1 Sum of Taylor Expansions
8.4.2 Product of Taylor Expansions
8.4.3 Quotient of Taylor Expansions
8.4.4 Taylor Expansion of a Composite Function
8.4.5 Integration of a Taylor Expansion
8.5 Generalized Taylor Expansions
8.6 Applications of Taylor Expansions
8.6.1 Calculation of Limits – Equivalent Functions
8.6.2 Local Behavior and Infinite Branches of Functions
8.7 Exercises with Solutions
Bibliography
Acheter

Livres dans la même catégorie

Avis des lecteurs

Aucun avis pour le moment.