A garden of integrals:
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Bibliographische Detailangaben
1. Verfasser: Burk, Frank (VerfasserIn)
Format: Elektronisch E-Book
Sprache:English
Veröffentlicht: Washington, DC Mathematical Association of America c2007
Schriftenreihe:Dolciani mathematical expositions no. 31
Schlagworte:
Online-Zugang:FAW01
FAW02
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Beschreibung: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
Beschreibung:1 Online-Ressource (xiv, 281 p.)
ISBN:088385337X
1614442096
9780883853375
9781614442097

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