Mathematical Analysis Fundamentals
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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.
Key Features
- 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
Table of Contents
- 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
Product details
- No. of pages: 362
- Language: English
- Copyright: © Elsevier 2014
- Published: March 3, 2014
- Imprint: Elsevier
- eBook ISBN: 9780128010501
- Paperback ISBN: 9780128102695
About the Author
Agamirza Bashirov
Affiliations and Expertise
Eastern Mediterranean University, Gazimagusa, Turkey
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