Garden of Integrals.
Burk, Frank E.
Garden of Integrals. - 1st ed. - 1 online resource (296 pages) - Dolciani Mathematical Expositions ; v.31 . - Dolciani Mathematical Expositions .
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 Lebesgue4.6 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 Lebesgue9.7 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.
9781614442097
Integrals.
Electronic books.
QA308 -- .B868 2007eb
515/.43
Garden of Integrals. - 1st ed. - 1 online resource (296 pages) - Dolciani Mathematical Expositions ; v.31 . - Dolciani Mathematical Expositions .
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 Lebesgue4.6 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 Lebesgue9.7 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.
9781614442097
Integrals.
Electronic books.
QA308 -- .B868 2007eb
515/.43