Mathematical analysis fundamentals
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...
| Otros Autores: | |
|---|---|
| Formato: | Libro electrónico |
| Idioma: | Inglés |
| Publicado: |
London, England ; Waltham, Massachusetts :
Elsevier
2014
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| Edición: | First edition |
| Colección: | Elsevier insights
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| Materias: | |
| Ver en Biblioteca Universitat Ramon Llull: | https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009629649506719 |
Tabla de Contenidos:
- Half Title; Title Page; Copyright; Dedication; Contents; Preface; Acknowledgments; 1 Sets and Proofs; 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; 2 Numbers; 2.1 System mathbb N; 2.2 Systems mathbb Z and mathbb Q; 2.3 Least Upper Bound Property and mathbb Q; 2.4 System mathbb R; 2.5 Least Upper Bound Property and mathbb R; 2.6* Systems textbar textmathbb R, mathbb C, and presup ast mathbb R; 2.7 Cardinality; 3 Convergence; 3.1 Convergence of Numerical Sequences
- 3.2 Cauchy Criterion for Convergence3.3 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; 4 Point Set Topology; 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; 5 Continuity; 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 -valued Functions6 Space C(E, E); 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; 7 Differentiation; 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 mathbb R; 8 Bounded Variation; 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; 9 Riemann Integration; 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; 10 Generalizations of Riemann Integration; 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
- 11 Transcendental Functions11.1 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; 12 Fourier Series and Integrals; 12.1 Trigonometric Series; 12.2 Riemann'226Lebesgue 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
