A garden of integrals /:
The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there are a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy,...
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| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Washington, DC :
Mathematical Association of America,
©2007.
|
| Series: | Dolciani mathematical expositions ;
no. 31. |
| Subjects: | |
| Online Access: | DE-862 DE-863 |
| Summary: | The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there are a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reason for their existence and their uses are given. Historical information is plentiful. Advanced undergraduate mathematics majors, graduate students, and faculty members are the audience for the book. Even experienced faculty members are unlikely to be aware of all of the integrals in the Garden of Integrals and the book provides an opportunity to see them and appreciate the richness of the idea of integral. Professor Burke's clear and well-motivated exposition makes this book a joy to read. |
| Physical Description: | 1 online resource (xiv, 281 pages) : illustrations |
| Bibliography: | Includes bibliographical references and index. |
| ISBN: | 9781614442097 1614442096 |
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| 100 | 1 | |a Burk, Frank. | |
| 245 | 1 | 2 | |a A garden of integrals / |c Frank Burk. |
| 260 | |a Washington, DC : |b Mathematical Association of America, |c ©2007. | ||
| 300 | |a 1 online resource (xiv, 281 pages) : |b illustrations | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 490 | 1 | |a Dolciani mathematical expositions ; |v no. 31 | |
| 504 | |a Includes bibliographical references and index. | ||
| 505 | 0 | 0 | |t Foreword -- |g An historical overview -- |g 1.1. |t Rearrangements -- |g 1.2. |t The lune of Hippocrates -- |g 1.3. |t Exdoxus and the method of exhaustion -- |g 1.4. |t Archimedes' method -- |t 1.5. |t Gottfried Leibniz and Isaac Newton -- |g 1.6. |t Augustin-Louis Cauchy -- |g 1.7. |t Bernhard Riemann -- |g 1.8. |t Thomas Stieltjes -- |g 1.9. |t Henri Lebesgue -- |g 1.10. |t The Lebesgue-Stieltjes integral -- |g 1.11. |t Ralph Henstock and Jaroslav Kurzweil -- |g 1.12. |t Norbert Wiener -- |g 1.13. |t Richard Feynman -- |g 1.14. |t References -- |g 2. |t The Cauchy integral -- |g 2.1. |t Exploring integration -- |g 2.2. |t Cauchy's integral -- |g 2.3. |t Recovering functions by integration -- |g 2.4. |t Recovering functions by differentiation -- |g 2.5. |t A convergence theorem -- |g 2.6. |t Joseph Fourier -- |g 2.7. |t P.G. Lejeune Dirichlet -- |g 2.8. |t Patrick Billingsley's example -- |g 2.9. |t Summary -- |g 2.10. |t References -- |g 3. |t The Riemann integral -- |g 3.1. |t Riemann's integral -- |g 3.2. |t Criteria for Riemann integrability -- |g 3.3. |t Cauchy and Darboux criteria for Riemann integrability -- |g 3.4. |t Weakening continuity -- |g 3.5. |t Monotonic functions are Riemann integrable -- |g 3.6. |t Lebesgue's criteria -- |g 3.7. |t Evaluating à la Riemann -- |g 3.8. |t Sequences of Riemann integrable functions -- |g 3.9. |t The Cantor set -- |g 3.10. |t A nowhere dense set of positive measure -- |g 3.11. |t Cantor functions -- |g 3.12. |t Volterra's example -- |g 3.13. |t Lengths of graphs and the Cantor function -- |g 3.14. |t Summary -- |g 3.15. |t References. |
| 505 | 0 | 0 | |g 4. |t Riemann-Stieltjes integral -- |g 4.1. |t Generalizing the Riemann integral-- |g 4.2. |t Discontinuities -- |g 4.3. |t Existence of Riemann-Stieltjes integrals -- |g 4.4. |t Monotonicity of [null] -- |g 4.5. |t Euler's summation formula -- |g 4.6. |t Uniform convergence and R-S integration -- |g 4.7. |t References -- |g 5. |t Lebesgue measure -- |g 5.1. |t Lebesgue's idea -- |g 5.2. |t Measurable sets -- |g 5.3. |t Lebesgue measurable sets and Carathéodory -- |g 5.4. |t Sigma algebras -- |g 5.5. |t Borel sets -- |g 5.6. |t Approximating measurable sets -- |g 5.7. |t Measurable functions -- |g 5.8. |t More measureable functions -- |g 5.9. |t What does monotonicity tell us? -- |g 5.10. |t Lebesgue's differentiation theorem -- |g 5.11. |t References -- |g 6. |t The Lebesgue-Stieltjes integral -- |g 6.1. |t Introduction -- |g 6.2. |t Integrability : Riemann ensures Lebesgue -- |g 6.3. |t Convergence theorems -- |g 6.4. |t Fundamental theorems for the Lebesgue integral -- |g 6.5. |t Spaces -- |g 6.6. |t L²[-pi, pi] and Fourier series -- |g 6.7. |t Lebesgue measure in the plane and Fubini's theorem -- |g 6.8. |t Summary-- |t References -- |g 7. |t The Lebesgue-Stieltjes integral -- |g 7.1. |t L-S measures and monotone increasing functions -- |g 7.2. |t Carathéodory's measurability criterion -- |g 7.3. |t Avoiding complacency -- |g 7.4. |t L-S measures and nonnegative Lebesgue integrable functions -- |g 7.5. |t L-S measures and random variables -- |g 7.6. |t The Lebesgue-Stieltjes integral -- |g 7.7. |t A fundamental theorem for L-S integrals -- |g 7.8. |t References. |
| 505 | 0 | 0 | |g 8. |t The Henstock-Kurzweil integral -- |g 8.1. |t The generalized Riemann integral -- |g 8.2. |t Gauges and [infinity]-fine partitions -- |g 8.3. |t H-K integrable functions -- |g 8.4. |t The Cauchy criterion for H-K integrability -- |g 8.5. |t Henstock's lemma -- |g 8.6. |t Convergence theorems for the H-K integral -- |g 8.7. |t Some properties of the H-K integral -- |g 8.8. |t The second fundamental theorem -- |g 8.9. |t Summary-- |g 8.10. |t References -- |g 9. |t The Wiener integral -- |g 9.1. |t Brownian motion -- |g 9.2. |t Construction of the Wiener measure -- |g 9.3. |t Wiener's theorem -- |g 9.4. |t Measurable functionals -- |g 9.5. |t The Wiener integral -- |g 9.6. |t Functionals dependent on a finite number of t values -- |g 9.7. |t Kac's theorem -- |g 9.8. |t References -- |g 10. |t Feynman integral -- |g 10.1. |t Introduction -- |g 10.2. |t Summing probability amplitudes -- |g 10.3. |t A simple example -- |g 10.4. |t The Fourier transform -- |g 10.5. |t The convolution product -- |g 10.6. |t The Schwartz space -- |g 10.7. |t Solving Schrödinger problem A -- |g 10.8. |t An abstract Cauchy problem -- |g 10.9. |t Solving in the Schwartz space -- |g 10.10. |t Solving Schrödinger problem B -- |g 10.11. |t References -- |t Index -- |t About the author. |
| 588 | 0 | |a Print version record. | |
| 520 | |a The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there are a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reason for their existence and their uses are given. Historical information is plentiful. Advanced undergraduate mathematics majors, graduate students, and faculty members are the audience for the book. Even experienced faculty members are unlikely to be aware of all of the integrals in the Garden of Integrals and the book provides an opportunity to see them and appreciate the richness of the idea of integral. Professor Burke's clear and well-motivated exposition makes this book a joy to read. | ||
| 546 | |a English. | ||
| 650 | 0 | |a Integrals. |0 http://id.loc.gov/authorities/subjects/sh85067099 | |
| 650 | 6 | |a Intégrales. | |
| 650 | 7 | |a MATHEMATICS |x Calculus. |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS |x Mathematical Analysis. |2 bisacsh | |
| 650 | 7 | |a Integrals |2 fast | |
| 650 | 7 | |a Reelle Funktion |2 gnd |0 http://d-nb.info/gnd/4048918-8 | |
| 650 | 7 | |a Integral |2 gnd |0 http://d-nb.info/gnd/4131477-3 | |
| 700 | 1 | |a Scully, Terence, |d 1935- |1 https://id.oclc.org/worldcat/entity/E39PBJpJcMWTVg6bFCK8hwgdwC |0 http://id.loc.gov/authorities/names/n82132823 | |
| 776 | 0 | 8 | |i Print version: |a Burk, Frank. |t Garden of integrals. |d Washington, DC : Mathematical Association of America, ©2007 |z 9780883853375 |w (DLC) 2007925414 |w (OCoLC)156995360 |
| 830 | 0 | |a Dolciani mathematical expositions ; |v no. 31. |0 http://id.loc.gov/authorities/names/n42009859 | |
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| contents | Foreword -- Rearrangements -- The lune of Hippocrates -- Exdoxus and the method of exhaustion -- Archimedes' method -- 1.5. Gottfried Leibniz and Isaac Newton -- Augustin-Louis Cauchy -- Bernhard Riemann -- Thomas Stieltjes -- Henri Lebesgue -- The Lebesgue-Stieltjes integral -- Ralph Henstock and Jaroslav Kurzweil -- Norbert Wiener -- Richard Feynman -- References -- The Cauchy integral -- Exploring integration -- Cauchy's integral -- Recovering functions by integration -- Recovering functions by differentiation -- A convergence theorem -- Joseph Fourier -- P.G. Lejeune Dirichlet -- Patrick Billingsley's example -- Summary -- The Riemann integral -- Riemann's integral -- Criteria for Riemann integrability -- Cauchy and Darboux criteria for Riemann integrability -- Weakening continuity -- Monotonic functions are Riemann integrable -- Lebesgue's criteria -- Evaluating à la Riemann -- Sequences of Riemann integrable functions -- The Cantor set -- A nowhere dense set of positive measure -- Cantor functions -- Volterra's example -- Lengths of graphs and the Cantor function -- References. Riemann-Stieltjes integral -- Generalizing the Riemann integral-- Discontinuities -- Existence of Riemann-Stieltjes integrals -- Monotonicity of [null] -- Euler's summation formula -- Uniform convergence and R-S integration -- Lebesgue measure -- Lebesgue's idea -- Measurable sets -- Lebesgue measurable sets and Carathéodory -- Sigma algebras -- Borel sets -- Approximating measurable sets -- Measurable functions -- More measureable functions -- What does monotonicity tell us? -- Lebesgue's differentiation theorem -- Introduction -- Integrability : Riemann ensures Lebesgue -- Convergence theorems -- Fundamental theorems for the Lebesgue integral -- Spaces -- L²[-pi, pi] and Fourier series -- Lebesgue measure in the plane and Fubini's theorem -- Summary-- L-S measures and monotone increasing functions -- Carathéodory's measurability criterion -- Avoiding complacency -- L-S measures and nonnegative Lebesgue integrable functions -- L-S measures and random variables -- A fundamental theorem for L-S integrals -- The Henstock-Kurzweil integral -- The generalized Riemann integral -- Gauges and [infinity]-fine partitions -- H-K integrable functions -- The Cauchy criterion for H-K integrability -- Henstock's lemma -- Convergence theorems for the H-K integral -- Some properties of the H-K integral -- The second fundamental theorem -- The Wiener integral -- Brownian motion -- Construction of the Wiener measure -- Wiener's theorem -- Measurable functionals -- Functionals dependent on a finite number of t values -- Kac's theorem -- Feynman integral -- Summing probability amplitudes -- A simple example -- The Fourier transform -- The convolution product -- The Schwartz space -- Solving Schrödinger problem A -- An abstract Cauchy problem -- Solving in the Schwartz space -- Solving Schrödinger problem B -- Index -- About the author. |
| ctrlnum | (OCoLC)793207766 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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--</subfield><subfield code="g">1.11.</subfield><subfield code="t">Ralph Henstock and Jaroslav Kurzweil --</subfield><subfield code="g">1.12.</subfield><subfield code="t">Norbert Wiener --</subfield><subfield code="g">1.13.</subfield><subfield code="t">Richard Feynman --</subfield><subfield code="g">1.14.</subfield><subfield code="t">References --</subfield><subfield code="g">2.</subfield><subfield code="t">The Cauchy integral --</subfield><subfield code="g">2.1.</subfield><subfield code="t">Exploring integration --</subfield><subfield code="g">2.2.</subfield><subfield code="t">Cauchy's integral --</subfield><subfield code="g">2.3.</subfield><subfield code="t">Recovering functions by integration --</subfield><subfield code="g">2.4.</subfield><subfield code="t">Recovering functions by differentiation --</subfield><subfield code="g">2.5.</subfield><subfield code="t">A convergence theorem --</subfield><subfield code="g">2.6.</subfield><subfield code="t">Joseph Fourier --</subfield><subfield code="g">2.7.</subfield><subfield code="t">P.G. Lejeune Dirichlet --</subfield><subfield code="g">2.8.</subfield><subfield code="t">Patrick Billingsley's example --</subfield><subfield code="g">2.9.</subfield><subfield code="t">Summary --</subfield><subfield code="g">2.10.</subfield><subfield code="t">References --</subfield><subfield code="g">3.</subfield><subfield code="t">The Riemann integral --</subfield><subfield code="g">3.1.</subfield><subfield code="t">Riemann's integral --</subfield><subfield code="g">3.2.</subfield><subfield code="t">Criteria for Riemann integrability --</subfield><subfield code="g">3.3.</subfield><subfield code="t">Cauchy and Darboux criteria for Riemann integrability --</subfield><subfield code="g">3.4.</subfield><subfield code="t">Weakening continuity --</subfield><subfield code="g">3.5.</subfield><subfield code="t">Monotonic functions are Riemann integrable --</subfield><subfield code="g">3.6.</subfield><subfield code="t">Lebesgue's criteria --</subfield><subfield code="g">3.7.</subfield><subfield code="t">Evaluating à la Riemann --</subfield><subfield code="g">3.8.</subfield><subfield code="t">Sequences of Riemann integrable functions --</subfield><subfield code="g">3.9.</subfield><subfield code="t">The Cantor set --</subfield><subfield code="g">3.10.</subfield><subfield code="t">A nowhere dense set of positive measure --</subfield><subfield code="g">3.11.</subfield><subfield code="t">Cantor functions --</subfield><subfield code="g">3.12.</subfield><subfield code="t">Volterra's example --</subfield><subfield code="g">3.13.</subfield><subfield code="t">Lengths of graphs and the Cantor function --</subfield><subfield code="g">3.14.</subfield><subfield code="t">Summary --</subfield><subfield code="g">3.15.</subfield><subfield code="t">References.</subfield></datafield><datafield tag="505" ind1="0" ind2="0"><subfield code="g">4.</subfield><subfield code="t">Riemann-Stieltjes integral --</subfield><subfield code="g">4.1.</subfield><subfield code="t">Generalizing the Riemann integral--</subfield><subfield code="g">4.2.</subfield><subfield code="t">Discontinuities --</subfield><subfield code="g">4.3.</subfield><subfield code="t">Existence of Riemann-Stieltjes integrals --</subfield><subfield code="g">4.4.</subfield><subfield code="t">Monotonicity of [null] --</subfield><subfield code="g">4.5.</subfield><subfield code="t">Euler's summation formula --</subfield><subfield code="g">4.6.</subfield><subfield code="t">Uniform convergence and R-S integration --</subfield><subfield code="g">4.7.</subfield><subfield code="t">References --</subfield><subfield code="g">5.</subfield><subfield code="t">Lebesgue measure --</subfield><subfield code="g">5.1.</subfield><subfield code="t">Lebesgue's idea --</subfield><subfield code="g">5.2.</subfield><subfield code="t">Measurable sets --</subfield><subfield code="g">5.3.</subfield><subfield code="t">Lebesgue measurable sets and Carathéodory --</subfield><subfield code="g">5.4.</subfield><subfield code="t">Sigma algebras --</subfield><subfield code="g">5.5.</subfield><subfield code="t">Borel sets --</subfield><subfield code="g">5.6.</subfield><subfield code="t">Approximating measurable sets --</subfield><subfield code="g">5.7.</subfield><subfield code="t">Measurable functions --</subfield><subfield code="g">5.8.</subfield><subfield code="t">More measureable functions --</subfield><subfield code="g">5.9.</subfield><subfield code="t">What does monotonicity tell us? --</subfield><subfield code="g">5.10.</subfield><subfield code="t">Lebesgue's differentiation theorem --</subfield><subfield code="g">5.11.</subfield><subfield code="t">References --</subfield><subfield code="g">6.</subfield><subfield code="t">The Lebesgue-Stieltjes integral --</subfield><subfield code="g">6.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">6.2.</subfield><subfield code="t">Integrability : Riemann ensures Lebesgue --</subfield><subfield code="g">6.3.</subfield><subfield code="t">Convergence theorems --</subfield><subfield code="g">6.4.</subfield><subfield code="t">Fundamental theorems for the Lebesgue integral --</subfield><subfield code="g">6.5.</subfield><subfield code="t">Spaces --</subfield><subfield code="g">6.6.</subfield><subfield code="t">L²[-pi, pi] and Fourier series --</subfield><subfield code="g">6.7.</subfield><subfield code="t">Lebesgue measure in the plane and Fubini's theorem --</subfield><subfield code="g">6.8.</subfield><subfield code="t">Summary--</subfield><subfield code="t">References --</subfield><subfield code="g">7.</subfield><subfield code="t">The Lebesgue-Stieltjes integral --</subfield><subfield code="g">7.1.</subfield><subfield code="t">L-S measures and monotone increasing functions --</subfield><subfield code="g">7.2.</subfield><subfield code="t">Carathéodory's measurability criterion --</subfield><subfield code="g">7.3.</subfield><subfield code="t">Avoiding complacency --</subfield><subfield code="g">7.4.</subfield><subfield code="t">L-S measures and nonnegative Lebesgue integrable functions --</subfield><subfield code="g">7.5.</subfield><subfield code="t">L-S measures and random variables --</subfield><subfield code="g">7.6.</subfield><subfield code="t">The Lebesgue-Stieltjes integral --</subfield><subfield code="g">7.7.</subfield><subfield code="t">A fundamental theorem for L-S integrals --</subfield><subfield code="g">7.8.</subfield><subfield code="t">References.</subfield></datafield><datafield tag="505" ind1="0" ind2="0"><subfield code="g">8.</subfield><subfield code="t">The Henstock-Kurzweil integral --</subfield><subfield code="g">8.1.</subfield><subfield code="t">The generalized Riemann integral --</subfield><subfield code="g">8.2.</subfield><subfield code="t">Gauges and [infinity]-fine partitions --</subfield><subfield code="g">8.3.</subfield><subfield code="t">H-K integrable functions --</subfield><subfield code="g">8.4.</subfield><subfield code="t">The Cauchy criterion for H-K integrability --</subfield><subfield code="g">8.5.</subfield><subfield code="t">Henstock's lemma --</subfield><subfield code="g">8.6.</subfield><subfield code="t">Convergence theorems for the H-K integral --</subfield><subfield code="g">8.7.</subfield><subfield code="t">Some properties of the H-K integral --</subfield><subfield code="g">8.8.</subfield><subfield code="t">The second fundamental theorem --</subfield><subfield code="g">8.9.</subfield><subfield code="t">Summary--</subfield><subfield code="g">8.10.</subfield><subfield code="t">References --</subfield><subfield code="g">9.</subfield><subfield code="t">The Wiener integral --</subfield><subfield code="g">9.1.</subfield><subfield code="t">Brownian motion --</subfield><subfield code="g">9.2.</subfield><subfield code="t">Construction of the Wiener measure --</subfield><subfield code="g">9.3.</subfield><subfield code="t">Wiener's theorem --</subfield><subfield code="g">9.4.</subfield><subfield code="t">Measurable functionals --</subfield><subfield code="g">9.5.</subfield><subfield code="t">The Wiener integral --</subfield><subfield code="g">9.6.</subfield><subfield code="t">Functionals dependent on a finite number of t values --</subfield><subfield code="g">9.7.</subfield><subfield code="t">Kac's theorem --</subfield><subfield code="g">9.8.</subfield><subfield code="t">References --</subfield><subfield code="g">10.</subfield><subfield code="t">Feynman integral --</subfield><subfield code="g">10.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">10.2.</subfield><subfield code="t">Summing probability amplitudes --</subfield><subfield code="g">10.3.</subfield><subfield code="t">A simple example --</subfield><subfield code="g">10.4.</subfield><subfield code="t">The Fourier transform --</subfield><subfield code="g">10.5.</subfield><subfield code="t">The convolution product --</subfield><subfield code="g">10.6.</subfield><subfield code="t">The Schwartz space --</subfield><subfield code="g">10.7.</subfield><subfield code="t">Solving Schrödinger problem A --</subfield><subfield code="g">10.8.</subfield><subfield code="t">An abstract Cauchy problem --</subfield><subfield code="g">10.9.</subfield><subfield code="t">Solving in the Schwartz space --</subfield><subfield code="g">10.10.</subfield><subfield code="t">Solving Schrödinger problem B --</subfield><subfield code="g">10.11.</subfield><subfield code="t">References --</subfield><subfield code="t">Index --</subfield><subfield code="t">About the author.</subfield></datafield><datafield tag="588" ind1="0" ind2=" "><subfield code="a">Print version record.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there are a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reason for their existence and their uses are given. Historical information is plentiful. Advanced undergraduate mathematics majors, graduate students, and faculty members are the audience for the book. Even experienced faculty members are unlikely to be aware of all of the integrals in the Garden of Integrals and the book provides an opportunity to see them and appreciate the richness of the idea of integral. Professor Burke's clear and well-motivated exposition makes this book a joy to read.</subfield></datafield><datafield tag="546" ind1=" " ind2=" "><subfield code="a">English.</subfield></datafield><datafield tag="650" ind1=" " ind2="0"><subfield code="a">Integrals.</subfield><subfield code="0">http://id.loc.gov/authorities/subjects/sh85067099</subfield></datafield><datafield tag="650" ind1=" " ind2="6"><subfield code="a">Intégrales.</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">MATHEMATICS</subfield><subfield code="x">Calculus.</subfield><subfield code="2">bisacsh</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">MATHEMATICS</subfield><subfield code="x">Mathematical Analysis.</subfield><subfield code="2">bisacsh</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Integrals</subfield><subfield code="2">fast</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Reelle Funktion</subfield><subfield code="2">gnd</subfield><subfield code="0">http://d-nb.info/gnd/4048918-8</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Integral</subfield><subfield code="2">gnd</subfield><subfield code="0">http://d-nb.info/gnd/4131477-3</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Scully, Terence,</subfield><subfield code="d">1935-</subfield><subfield code="1">https://id.oclc.org/worldcat/entity/E39PBJpJcMWTVg6bFCK8hwgdwC</subfield><subfield code="0">http://id.loc.gov/authorities/names/n82132823</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Print version:</subfield><subfield code="a">Burk, Frank.</subfield><subfield code="t">Garden of integrals.</subfield><subfield code="d">Washington, DC : Mathematical Association of America, ©2007</subfield><subfield code="z">9780883853375</subfield><subfield code="w">(DLC) 2007925414</subfield><subfield code="w">(OCoLC)156995360</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Dolciani mathematical expositions ;</subfield><subfield code="v">no. 31.</subfield><subfield code="0">http://id.loc.gov/authorities/names/n42009859</subfield></datafield><datafield tag="966" ind1="4" ind2="0"><subfield code="l">DE-862</subfield><subfield code="p">ZDB-4-EBA</subfield><subfield code="q">FWS_PDA_EBA</subfield><subfield code="u">https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=450282</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="966" ind1="4" ind2="0"><subfield code="l">DE-863</subfield><subfield code="p">ZDB-4-EBA</subfield><subfield code="q">FWS_PDA_EBA</subfield><subfield code="u">https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=450282</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">EBL - Ebook Library</subfield><subfield code="b">EBLB</subfield><subfield code="n">EBL3330374</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">ebrary</subfield><subfield code="b">EBRY</subfield><subfield code="n">ebr10728523</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">EBSCOhost</subfield><subfield code="b">EBSC</subfield><subfield code="n">450282</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">YBP Library Services</subfield><subfield code="b">YANK</subfield><subfield code="n">7648349</subfield></datafield><datafield tag="938" ind1=" " ind2=" "><subfield code="a">Internet Archive</subfield><subfield code="b">INAR</subfield><subfield code="n">gardenofintegral0000burk</subfield></datafield><datafield tag="994" ind1=" " ind2=" "><subfield code="a">92</subfield><subfield code="b">GEBAY</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">ZDB-4-EBA</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-862</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-863</subfield></datafield></record></collection> |
| id | ZDB-4-EBA-ocn793207766 |
| illustrated | Illustrated |
| indexdate | 2025-04-11T08:37:41Z |
| institution | BVB |
| isbn | 9781614442097 1614442096 |
| language | English |
| lccn | 2007925414 |
| oclc_num | 793207766 |
| open_access_boolean | |
| owner | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| physical | 1 online resource (xiv, 281 pages) : illustrations |
| psigel | ZDB-4-EBA FWS_PDA_EBA ZDB-4-EBA |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Mathematical Association of America, |
| record_format | marc |
| series | Dolciani mathematical expositions ; |
| series2 | Dolciani mathematical expositions ; |
| spelling | Burk, Frank. A garden of integrals / Frank Burk. Washington, DC : Mathematical Association of America, ©2007. 1 online resource (xiv, 281 pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier Dolciani mathematical expositions ; no. 31 Includes bibliographical references and index. 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. 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. 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. Print version record. The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there are a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reason for their existence and their uses are given. Historical information is plentiful. Advanced undergraduate mathematics majors, graduate students, and faculty members are the audience for the book. Even experienced faculty members are unlikely to be aware of all of the integrals in the Garden of Integrals and the book provides an opportunity to see them and appreciate the richness of the idea of integral. Professor Burke's clear and well-motivated exposition makes this book a joy to read. English. Integrals. http://id.loc.gov/authorities/subjects/sh85067099 Intégrales. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Integrals fast Reelle Funktion gnd http://d-nb.info/gnd/4048918-8 Integral gnd http://d-nb.info/gnd/4131477-3 Scully, Terence, 1935- https://id.oclc.org/worldcat/entity/E39PBJpJcMWTVg6bFCK8hwgdwC http://id.loc.gov/authorities/names/n82132823 Print version: Burk, Frank. Garden of integrals. Washington, DC : Mathematical Association of America, ©2007 9780883853375 (DLC) 2007925414 (OCoLC)156995360 Dolciani mathematical expositions ; no. 31. http://id.loc.gov/authorities/names/n42009859 |
| spellingShingle | Burk, Frank A garden of integrals / Dolciani mathematical expositions ; Foreword -- Rearrangements -- The lune of Hippocrates -- Exdoxus and the method of exhaustion -- Archimedes' method -- 1.5. Gottfried Leibniz and Isaac Newton -- Augustin-Louis Cauchy -- Bernhard Riemann -- Thomas Stieltjes -- Henri Lebesgue -- The Lebesgue-Stieltjes integral -- Ralph Henstock and Jaroslav Kurzweil -- Norbert Wiener -- Richard Feynman -- References -- The Cauchy integral -- Exploring integration -- Cauchy's integral -- Recovering functions by integration -- Recovering functions by differentiation -- A convergence theorem -- Joseph Fourier -- P.G. Lejeune Dirichlet -- Patrick Billingsley's example -- Summary -- The Riemann integral -- Riemann's integral -- Criteria for Riemann integrability -- Cauchy and Darboux criteria for Riemann integrability -- Weakening continuity -- Monotonic functions are Riemann integrable -- Lebesgue's criteria -- Evaluating à la Riemann -- Sequences of Riemann integrable functions -- The Cantor set -- A nowhere dense set of positive measure -- Cantor functions -- Volterra's example -- Lengths of graphs and the Cantor function -- References. Riemann-Stieltjes integral -- Generalizing the Riemann integral-- Discontinuities -- Existence of Riemann-Stieltjes integrals -- Monotonicity of [null] -- Euler's summation formula -- Uniform convergence and R-S integration -- Lebesgue measure -- Lebesgue's idea -- Measurable sets -- Lebesgue measurable sets and Carathéodory -- Sigma algebras -- Borel sets -- Approximating measurable sets -- Measurable functions -- More measureable functions -- What does monotonicity tell us? -- Lebesgue's differentiation theorem -- Introduction -- Integrability : Riemann ensures Lebesgue -- Convergence theorems -- Fundamental theorems for the Lebesgue integral -- Spaces -- L²[-pi, pi] and Fourier series -- Lebesgue measure in the plane and Fubini's theorem -- Summary-- L-S measures and monotone increasing functions -- Carathéodory's measurability criterion -- Avoiding complacency -- L-S measures and nonnegative Lebesgue integrable functions -- L-S measures and random variables -- A fundamental theorem for L-S integrals -- The Henstock-Kurzweil integral -- The generalized Riemann integral -- Gauges and [infinity]-fine partitions -- H-K integrable functions -- The Cauchy criterion for H-K integrability -- Henstock's lemma -- Convergence theorems for the H-K integral -- Some properties of the H-K integral -- The second fundamental theorem -- The Wiener integral -- Brownian motion -- Construction of the Wiener measure -- Wiener's theorem -- Measurable functionals -- Functionals dependent on a finite number of t values -- Kac's theorem -- Feynman integral -- Summing probability amplitudes -- A simple example -- The Fourier transform -- The convolution product -- The Schwartz space -- Solving Schrödinger problem A -- An abstract Cauchy problem -- Solving in the Schwartz space -- Solving Schrödinger problem B -- Index -- About the author. Integrals. http://id.loc.gov/authorities/subjects/sh85067099 Intégrales. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Integrals fast Reelle Funktion gnd http://d-nb.info/gnd/4048918-8 Integral gnd http://d-nb.info/gnd/4131477-3 |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85067099 http://d-nb.info/gnd/4048918-8 http://d-nb.info/gnd/4131477-3 |
| title | A garden of integrals / |
| title_alt | Foreword -- Rearrangements -- The lune of Hippocrates -- Exdoxus and the method of exhaustion -- Archimedes' method -- 1.5. Gottfried Leibniz and Isaac Newton -- Augustin-Louis Cauchy -- Bernhard Riemann -- Thomas Stieltjes -- Henri Lebesgue -- The Lebesgue-Stieltjes integral -- Ralph Henstock and Jaroslav Kurzweil -- Norbert Wiener -- Richard Feynman -- References -- The Cauchy integral -- Exploring integration -- Cauchy's integral -- Recovering functions by integration -- Recovering functions by differentiation -- A convergence theorem -- Joseph Fourier -- P.G. Lejeune Dirichlet -- Patrick Billingsley's example -- Summary -- The Riemann integral -- Riemann's integral -- Criteria for Riemann integrability -- Cauchy and Darboux criteria for Riemann integrability -- Weakening continuity -- Monotonic functions are Riemann integrable -- Lebesgue's criteria -- Evaluating à la Riemann -- Sequences of Riemann integrable functions -- The Cantor set -- A nowhere dense set of positive measure -- Cantor functions -- Volterra's example -- Lengths of graphs and the Cantor function -- References. Riemann-Stieltjes integral -- Generalizing the Riemann integral-- Discontinuities -- Existence of Riemann-Stieltjes integrals -- Monotonicity of [null] -- Euler's summation formula -- Uniform convergence and R-S integration -- Lebesgue measure -- Lebesgue's idea -- Measurable sets -- Lebesgue measurable sets and Carathéodory -- Sigma algebras -- Borel sets -- Approximating measurable sets -- Measurable functions -- More measureable functions -- What does monotonicity tell us? -- Lebesgue's differentiation theorem -- Introduction -- Integrability : Riemann ensures Lebesgue -- Convergence theorems -- Fundamental theorems for the Lebesgue integral -- Spaces -- L²[-pi, pi] and Fourier series -- Lebesgue measure in the plane and Fubini's theorem -- Summary-- L-S measures and monotone increasing functions -- Carathéodory's measurability criterion -- Avoiding complacency -- L-S measures and nonnegative Lebesgue integrable functions -- L-S measures and random variables -- A fundamental theorem for L-S integrals -- The Henstock-Kurzweil integral -- The generalized Riemann integral -- Gauges and [infinity]-fine partitions -- H-K integrable functions -- The Cauchy criterion for H-K integrability -- Henstock's lemma -- Convergence theorems for the H-K integral -- Some properties of the H-K integral -- The second fundamental theorem -- The Wiener integral -- Brownian motion -- Construction of the Wiener measure -- Wiener's theorem -- Measurable functionals -- Functionals dependent on a finite number of t values -- Kac's theorem -- Feynman integral -- Summing probability amplitudes -- A simple example -- The Fourier transform -- The convolution product -- The Schwartz space -- Solving Schrödinger problem A -- An abstract Cauchy problem -- Solving in the Schwartz space -- Solving Schrödinger problem B -- Index -- About the author. |
| title_auth | A garden of integrals / |
| title_exact_search | A garden of integrals / |
| title_full | A garden of integrals / Frank Burk. |
| title_fullStr | A garden of integrals / Frank Burk. |
| title_full_unstemmed | A garden of integrals / Frank Burk. |
| title_short | A garden of integrals / |
| title_sort | garden of integrals |
| topic | Integrals. http://id.loc.gov/authorities/subjects/sh85067099 Intégrales. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Integrals fast Reelle Funktion gnd http://d-nb.info/gnd/4048918-8 Integral gnd http://d-nb.info/gnd/4131477-3 |
| topic_facet | Integrals. Intégrales. MATHEMATICS Calculus. MATHEMATICS Mathematical Analysis. Integrals Reelle Funktion Integral |
| work_keys_str_mv | AT burkfrank agardenofintegrals AT scullyterence agardenofintegrals AT burkfrank gardenofintegrals AT scullyterence gardenofintegrals |