简介
Though there is essentially only one derivative, there is a variety of integrals. In this book the basic properties of each are proved, their similarities and differences are pointed out, and the reasons for their existence and their uses are given. There is no other book like it.
目录
Foreword
1 An Historical Overview
1.1 Rearrangements
1.2 The Lune of Hippocrates
1.3 Eudoxus and the Method of Exhaustion
1.4 Archimedes' Method
1.5 Gottfried Leibniz and Isaac Newton
1.6 Augustin-Louis Cauchy
1.7 Bernhard Riemann
1.8 Thomas Stieltjes
1.9 Henri Lebesgue
1.10 The Lebesgue-Stieltjes Integral
1.11 Ralph Henstock and Jaroslav Kurzweil
1.12 Norbert Wiener
1.13 Richard Feynman
1.14 References
2 The Cauehy Integral
2.1 Exploring Integration
2.2 Cauchy's Integral
2.3 Recovering Functions by Integration
2.4 Recovering Functions by Differentiation
2.5 A Convergence Theorem
2.6 Joseph Fourier
2.7 P.G. Lejeune Dirichlet
2.8 Patrick Billingsley's Example
2.9 Summary
2.10 References
3 The Riemann Integral
3.1 Riemann's Integral
3.2 Criteria for Riemann Integrability
3.3 Cauchy and Darboux Criteria for Riemann Integrability
3.4 Weakening Continuity
3.5 Monotonic Functions Are Riemann Integrable
3.6 Lebesgue's Criteria
3.7 Evaluating a la Riemann
3.8 Sequences of Riemann Integrable Functions
3.9 The Cantor Set (1883)
3.10 A Nowhere Dense Set of Positive Measure
3.11 Cantor Functions
3.12 Volterra's Example
3.13 Lengths of Graphs and the Cantor Function
3.14 Summary
3.15 References
4 The Riemann-Stieltjes Integral
4.1 Generalizing the Riemann Integral
4.2 Discontinuities
4.3 Existence of Riemann-Stieltjes Integrals
4.4 Monotonicity of
4.5 Euler's Summation Formula
4.6 Uniform Convergence and R-S Integration
4.7 References
5 Lebesgue Measure
5.1 Lebesgue's Idea
5.2 Measurable Sets
5.3 Lebesgue Measurable Sets and Carath~odory
5.4 Sigma Algebras
5.5 Borel Sets
5.6 Approximating Measurable Sets
5.7 Measurable Functions
5.8 More Measureable Functions
5.9 What Does Monotonicity Tell Us?
5.10 Lebesgue's Differentiation Theorem
5.11 References
6 The Lebesgue Integral
7 The Lebestue-Stieltjes Integral
8 The Henstock-Kurzweil Imtegral
9 The Wiener Integral
10 The Feynman Integral
Index
About the Author
1 An Historical Overview
1.1 Rearrangements
1.2 The Lune of Hippocrates
1.3 Eudoxus and the Method of Exhaustion
1.4 Archimedes' Method
1.5 Gottfried Leibniz and Isaac Newton
1.6 Augustin-Louis Cauchy
1.7 Bernhard Riemann
1.8 Thomas Stieltjes
1.9 Henri Lebesgue
1.10 The Lebesgue-Stieltjes Integral
1.11 Ralph Henstock and Jaroslav Kurzweil
1.12 Norbert Wiener
1.13 Richard Feynman
1.14 References
2 The Cauehy Integral
2.1 Exploring Integration
2.2 Cauchy's Integral
2.3 Recovering Functions by Integration
2.4 Recovering Functions by Differentiation
2.5 A Convergence Theorem
2.6 Joseph Fourier
2.7 P.G. Lejeune Dirichlet
2.8 Patrick Billingsley's Example
2.9 Summary
2.10 References
3 The Riemann Integral
3.1 Riemann's Integral
3.2 Criteria for Riemann Integrability
3.3 Cauchy and Darboux Criteria for Riemann Integrability
3.4 Weakening Continuity
3.5 Monotonic Functions Are Riemann Integrable
3.6 Lebesgue's Criteria
3.7 Evaluating a la Riemann
3.8 Sequences of Riemann Integrable Functions
3.9 The Cantor Set (1883)
3.10 A Nowhere Dense Set of Positive Measure
3.11 Cantor Functions
3.12 Volterra's Example
3.13 Lengths of Graphs and the Cantor Function
3.14 Summary
3.15 References
4 The Riemann-Stieltjes Integral
4.1 Generalizing the Riemann Integral
4.2 Discontinuities
4.3 Existence of Riemann-Stieltjes Integrals
4.4 Monotonicity of
4.5 Euler's Summation Formula
4.6 Uniform Convergence and R-S Integration
4.7 References
5 Lebesgue Measure
5.1 Lebesgue's Idea
5.2 Measurable Sets
5.3 Lebesgue Measurable Sets and Carath~odory
5.4 Sigma Algebras
5.5 Borel Sets
5.6 Approximating Measurable Sets
5.7 Measurable Functions
5.8 More Measureable Functions
5.9 What Does Monotonicity Tell Us?
5.10 Lebesgue's Differentiation Theorem
5.11 References
6 The Lebesgue Integral
7 The Lebestue-Stieltjes Integral
8 The Henstock-Kurzweil Imtegral
9 The Wiener Integral
10 The Feynman Integral
Index
About the Author
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