Mathematical Analysis Fundamentals
- 1st Edition - March 27, 2014
- Latest edition
- Author: Agamirza Bashirov
- Language: English
The author’s goal is a rigorous presentation of the fundamentals of analysis, starting from elementary level and moving to the advanced coursework. The curriculum of all ma… Read more
The author’s goal is a rigorous presentation of the fundamentals of analysis, starting from elementary level and moving to the advanced coursework. The curriculum of all mathematics (pure or applied) and physics programs include a compulsory course in mathematical analysis. This book will serve as can serve a main textbook of such (one semester) courses. The book can also serve as additional reading for such courses as real analysis, functional analysis, harmonic analysis etc. For non-math major students requiring math beyond calculus, this is a more friendly approach than many math-centric options.
- Friendly and well-rounded presentation of pre-analysis topics such as sets, proof techniques and systems of numbers
- Deeper discussion of the basic concept of convergence for the system of real numbers, pointing out its specific features, and for metric spaces
- Presentation of Riemann integration and its place in the whole integration theory for single variable, including the Kurzweil-Henstock integration
- Elements of multiplicative calculus aiming to demonstrate the non-absoluteness of Newtonian calculus
Undergraduates, students just beginning to study proof-based mathematics
- Dedication
- Preface
- Acknowledgments
- Chapter 1. Sets and Proofs
- Abstract
- 1.1 Sets, Elements, and Subsets
- 1.2 Operations on Sets
- 1.3 Language of Logic
- 1.4 Techniques of Proof
- 1.5 Relations
- 1.6 Functions
- 1.7* Axioms of Set Theory
- Exercises
- Chapter 2. Numbers
- Abstract
- 2.1 System N
- 2.2 Systems Z and Q
- 2.3 Least Upper Bound Property and Q
- 2.4 System R
- 2.5 Least Upper Bound Property and R
- 2.6* Systems R¯, C, and ∗R
- 2.7 Cardinality
- Exercises
- Chapter 3. Convergence
- Abstract
- 3.1 Convergence of Numerical Sequences
- 3.2 Cauchy Criterion for Convergence
- 3.3 Ordered Field Structure and Convergence
- 3.4 Subsequences
- 3.5 Numerical Series
- 3.6 Some Series of Particular Interest
- 3.7 Absolute Convergence
- 3.8 Number e
- Exercises
- Chapter 4. Point Set Topology
- Abstract
- 4.1 Metric Spaces
- 4.2 Open and Closed Sets
- 4.3 Completeness
- 4.4 Separability
- 4.5 Total Boundedness
- 4.6 Compactness
- 4.7 Perfectness
- 4.8 Connectedness
- 4.9* Structure of Open and Closed Sets
- Exercises
- Chapter 5. Continuity
- Abstract
- 5.1 Definition and Examples
- 5.2 Continuity and Limits
- 5.3 Continuity and Compactness
- 5.4 Continuity and Connectedness
- 5.5 Continuity and Oscillation
- 5.6 Continuity of Rk -valued Functions
- Exercises
- Chapter 6. Space C(E,E′)
- Abstract
- 6.1 Uniform Continuity
- 6.2 Uniform Convergence
- 6.3 Completeness of C(E,E′)
- 6.4 Bernstein and Weierstrass Theorems
- 6.5* Stone and Weierstrass Theorems
- 6.6* Ascoli-Arzelà Theorem
- Exercises
- Chapter 7. Differentiation
- Abstract
- 7.1 Derivative
- 7.2 Differentiation and Continuity
- 7.3 Rules of Differentiation
- 7.4 Mean-Value Theorems
- 7.5 Taylor’s Theorem
- 7.6* Differential Equations
- 7.7* Banach Spaces and the Space C1(a,b)
- 7.8 A View to Differentiation in Rk
- Exercises
- Chapter 8. Bounded Variation
- Abstract
- 8.1 Monotone Functions
- 8.2 Cantor Function
- 8.3 Functions of Bounded Variation
- 8.4 Space BV(a, b)
- 8.5 Continuous Functions of Bounded Variation
- 8.6 Rectifiable Curves
- Exercises
- Chapter 9. Riemann Integration
- Abstract
- 9.1 Definition of the Riemann Integral
- 9.2 Existence of the Riemann Integral
- 9.3 Lebesgue Characterization
- 9.4 Properties of the Riemann Integral
- 9.5 Riemann Integral Depending on a Parameter
- 9.6 Improper Integrals
- Exercises
- Chapter 10. Generalizations of Riemann Integration
- Abstract
- 10.1 Riemann– Stieltjes Integral
- 10.2* Helly’s Theorems
- 10.3* Reisz Representation
- 10.4* Definition of the Kurzweil– Henstock Integral
- 10.5* Differentiation of the Kurzweil– Henstock Integral
- 10.6* Lebesgue Integral
- Exercises
- Chapter 11. Transcendental Functions
- Abstract
- 11.1 Logarithmic and Exponential Functions
- 11.2* Multiplicative Calculus
- 11.3 Power Series
- 11.4 Analytic Functions
- 11.5 Hyperbolic and Trigonometric Functions
- 11.6 Infinite Products
- 11.7* Improper Integrals Depending on a Parameter
- 11.8* Euler’s Integrals
- Exercises
- Chapter 12. Fourier Series and Integrals
- Abstract
- 12.1 Trigonometric Series
- 12.2 Riemann–Lebesgue Lemma
- 12.3 Dirichlet Kernels and Riemann’s Localization Lemma
- 12.4 Pointwise Convergence of Fourier Series
- 12.5* Fourier Series in Inner Product Spaces
- 12.6* Cesàro Summability and Fejér’s Theorem
- 12.7 Uniform Convergence of Fourier Series
- 12.8* Gibbs Phenomenon
- 12.9* Fourier Integrals
- Exercises
- Bibliography
"...the book does have numerous good exercises that test the student’s grasp and cover some advanced topics, and it does give thorough coverage of much of classical analysis…"—MAA Reviews, 2014
- Edition: 1
- Latest edition
- Published: March 27, 2014
- Language: English
AB
Agamirza Bashirov
Affiliations and expertise
Eastern Mediterranean University, Gazimagusa, TurkeyRead Mathematical Analysis Fundamentals on ScienceDirect