Garden of Integrals.

Intro -- copyright page -- title page -- Foreword -- Contents -- 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 -- 1.5.1 Leibniz's Argument -- 1.5.2 Newton's Result -- 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 -- 2.1.1 Fermat's Formula -- 2.1.2 Wallis's Formula -- 2.1.3 Stirling's Formula -- 2.1.4 Stieltjes' Formula -- 2.2 Cauchy's Integral -- 2.2.1 Cauchy's Theorem (1823) -- 2.2.2 Cauchy Criteria for Cauchy Integrability -- 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 phi -- 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 -- 5.2.1 The Wish List and Lebesgue Outer Measure -- 5.3 Lebesgue Measurable Sets and Carathéodory -- 5.4 Sigma Algebras -- 5.5 Borel Sets -- 5.6 Approximating Measurable Sets -- 5.6.1 Vitali's Covering Theorem -- 5.7 Measurable Functions -- 5.7.1 Continuous Functions Defined on Measurable Sets -- 5.7.2 Riemann Integrable Functions -- 5.7.3 Limiting Operations and Measurability -- 5.7.4 Simple Functions -- 5.7.5 Pointwise Convergence Is Almost Uniform Convergence -- 5.8 More Measureable Functions -- 5.8.1 Functions of Bounded Variation -- 5.8.2 Functions of Bounded Variation and Monotone Functions -- 5.8.3 Absolutely Continuous Functions -- 5.9 What Does Monotonicity Tell Us? -- 5.9.1 Dini Derivates of a Function -- 5.10 Lebesgue's Differentiation Theorem -- 5.11 References -- 6 The Lebesgue Integral -- 6.1 Introduction -- 6.1.1 Lebesgue's Integral -- 6.1.2 Young's Approach -- 6.1.3 And Another Approach -- 6.2 Integrability: Riemann Ensures Lebesgue -- 6.2.1 Nonnegative Unbounded Measurable Functions -- 6.2.2 Positive and Negative Measurable Functions -- 6.2.3 Arbitrary Measurable Subsets -- 6.2.4 Another Definition of the Lebesgue Integral -- 6.3 Convergence Theorems -- 6.3.1 Monotone Convergence -- 6.3.2 Sequential Convergence -- 6.3.3 The Dominated Convergence Theorem -- 6.3.4 Interchanging summation and Integral -- 6.4 Fundamental Theorems for the Lebesgue Integral -- 6.4.1 Properties of the Indefinite Integral -- 6.4.2 A Fundamental Theorem for the Lebesgue Integral -- 6.4.3 The Other Fundamental Theorem -- 6.4.4 The Bounded Variation Condition -- 6.4.5 Another Fundamental Theorem of Calculus -- 6.4.6 Comments -- 6.5 Spaces -- 6.5.1 Metric Space -- 6.5.2 Famous Inequalities -- 6.5.3 Completeness.

6.5.4 The Riesz Completeness Theorem -- 6.6 L^2 [- pi, pi] and Fourier Series -- 6.7 Lebesgue Measure in the Plane and Fubini's Theorem -- 6.8 Summary -- 6.9 References -- 7 The Lebesgue{Stieltjes Integral -- 7.1 L-S Measures and Monotone Increasing Functions -- 7.1.1 Properties of the Weight Function -- 7.1.2 The L-S Outer Measure -- 7.2 Carathe odory's Measurability Criterion -- 7.3 Avoiding Complacency -- 7.4 L-S Measures and Nonnegative Lebesgue Integrable Functions -- 7.4.1 Properties of mu_f -- 7.5 L-S Measures and Random Variables -- 7.5.1 Properties of F_X -- 7.6 The Lebesgue{Stieltjes Integral -- 7.7 A Fundamental Theorem for L-S Integrals -- 7.8 Reference -- 8 The Henstock{Kurzweil Integral -- 8.1 The Generalized Riemann Integral -- 8.2 Gauges and delta-fine Partitions -- 8.3 H-K Integrable Functions -- 8.3.2 Riemann 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 -- 8.6.1 Monotone Convergence -- 8.6.2 Dominated Convergence -- 8.7 Some Properties of the H-K Integral -- 8.7.1 Extension of Lebesgue -- 8.7.2 Recovering Functions via Differentiation -- 8.7.3 H-K, but not Lebesgue, Integrable -- 8.7.4 Fundamental H-K Theorem -- 8.7.5 Sawtooth Functions -- 8.8 The Second Fundamental Theorem -- 8.8.1 Some H-K Properties -- 8.8.2 Second Fundamental Theorem for H-K -- 8.9 Summary -- 8.10 References -- 9 The Wiener Integral -- 9.1 Brownian Motion -- 9.1.1 Wiener's Explanation -- 9.2 Construction of the Wiener Measure -- 9.2.1 Borel Cylinders with One Restriction -- 9.2.2 Borel Cylinders with Two Restrictions -- 9.2.3 The Sigma Algebra B^[0.1] -- 9.3 Wiener's Theorem -- 9.3.1 Open Sets and Open Balls -- 9.4 Measurable Functionals -- 9.5 The Wiener Integral -- 9.6 Functionals Dependent on a Finite Number of t Values -- 9.6.1 More Interesting Functionals.

9.7 Kac's Theorem -- 9.8 References -- 10 The Feynman Integral -- 10.1 Introduction -- 10.1.1 Schrödinger's Equation -- 10.1.2 Feynman's Riemann Sums -- 10.2 Summing Probability Amplitudes -- 10.2.1 First Approximation -- 10.2.2 Second Approximation -- 10.2.3 The Normalizing Constant -- 10.3 A Simple Example -- 10.4 The Fourier Transform -- 10.5 The Convolution Product -- 10.6 The Schwartz Space -- 10.6.1 Plancherel's Theorem -- 10.7 Solving SchroŁ dinger Problem A -- 10.7.1 Finding the Right Space of Functions -- 10.8 An Abstract Cauchy Problem -- 10.8.1 Defining the Abstract Cauchy Problem -- 10.8.2 Operators on a Complex Hilbert Space -- 10.9 Solving in the Schwartz Space -- 10.9.1 Extending the Solution of Problem A -- 10.9.2 A Theorem -- 10.10 Solving Schrödinger Problem B -- 10.10.1 Prelude to Problem B -- 10.10.2 Trotter's Contribution -- 10.10.3 Semigroups of Linear Operators -- 10.10.4 Semigroup Terminology and a Theorem -- 10.10.5 Some Notes on Our Solution -- 10.10.6 Applying the Theorem -- 10.10.7 Problem B and the Trotter Product -- 10.11 References -- Index -- About the Author.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.