| Description |
1 online resource (xiv, 281 pages) : illustrations. |
| Series |
Dolciani mathematical expositions ; no. 31 |
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Dolciani mathematical expositions ; no. 31.
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| Bibliography |
Includes bibliographical references and index. |
| Contents |
Foreword -- An historical overview -- 1.1. Rearrangements -- 1.2. The lune of Hippocrates -- 1.3. Exdoxus 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 Cauchy 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 à la Riemann -- 3.8. Sequences of Riemann integrable functions -- 3.9. The Cantor set -- 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. |
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4. Riemann-Stieltjes integral -- 4.1. Generalizing the Riemann integral-- 4.2. Discontinuities -- 4.3. Existence of Riemann-Stieltjes integrals -- 4.4. Monotonicity of [null] -- 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 Carathéodory -- 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-Stieltjes integral -- 6.1. Introduction -- 6.2. Integrability : Riemann ensures Lebesgue -- 6.3. Convergence theorems -- 6.4. Fundamental theorems for the Lebesgue integral -- 6.5. Spaces -- 6.6. L²[-pi, pi] and Fourier series -- 6.7. Lebesgue measure in the plane and Fubini's theorem -- 6.8. Summary-- References -- 7. The Lebesgue-Stieltjes integral -- 7.1. L-S measures and monotone increasing functions -- 7.2. Carathéodory's measurability criterion -- 7.3. Avoiding complacency -- 7.4. L-S measures and nonnegative Lebesgue integrable functions -- 7.5. L-S measures and random variables -- 7.6. The Lebesgue-Stieltjes integral -- 7.7. A fundamental theorem for L-S integrals -- 7.8. References. |
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8. The Henstock-Kurzweil integral -- 8.1. The generalized Riemann integral -- 8.2. Gauges and [infinity]-fine partitions -- 8.3. H-K integrable functions -- 8.4. The Cauchy criterion for H-K integrability -- 8.5. Henstock's lemma -- 8.6. Convergence theorems for the H-K integral -- 8.7. Some properties of the H-K integral -- 8.8. The second fundamental theorem -- 8.9. Summary-- 8.10. References -- 9. The Wiener integral -- 9.1. Brownian motion -- 9.2. Construction of the Wiener measure -- 9.3. Wiener's theorem -- 9.4. Measurable functionals -- 9.5. The Wiener integral -- 9.6. Functionals dependent on a finite number of t values -- 9.7. Kac's theorem -- 9.8. References -- 10. Feynman integral -- 10.1. Introduction -- 10.2. Summing probability amplitudes -- 10.3. A simple example -- 10.4. The Fourier transform -- 10.5. The convolution product -- 10.6. The Schwartz space -- 10.7. Solving Schrödinger problem A -- 10.8. An abstract Cauchy problem -- 10.9. Solving in the Schwartz space -- 10.10. Solving Schrödinger problem B -- 10.11. References -- Index -- About the author. |
| Note |
Print version record. |
| Subject |
Integrals.
|
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MATHEMATICS -- Calculus.
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MATHEMATICS -- Mathematical Analysis.
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Integrals. (OCoLC)fst00975518
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| Genre/Form |
Electronic books.
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| Added Author |
Scully, Terence, 1935-
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| Other Form: |
Print version: Burk, Frank. Garden of integrals. Washington, DC : Mathematical Association of America, ©2007 9780883853375 (DLC) 2007925414 (OCoLC)156995360 |
| ISBN |
9781614442097 (electronic bk.) |
|
1614442096 (electronic bk.) |
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