作者介紹
(美)弗蘭克·E.伯克|責編:李鵬
弗蘭克·E.伯克
Frank Burk was raised in Liberty, Missouri, and graduated from Liberty HighSchool in 1960, a Merit Scholar. In 1963 he graduated from the University ofMissouri with a BA in mathematics. He earned his PhD in mathematics from theUniversity of California, Riverside in 1969.
Frank joined the faculty at California State University,Chico as a member ofthe mathematics department in 1968. He taught 37 years before retiring in 2004.He also taught for Butte College. Frank was the author of two books, this one andLebesgue Measure and Integration (Wiley,1998). He was a long-time member ofthe Mathematical Association of America.
Frank was a man of many interests. He was a runner his entire adult life andwas one of the founders of the Chico Running Club. Frank earned the rank ofEagle Scout in his youth and continued in scouting as scoutmaster of Troop 230 for18 years. Frank and his wife were foster parents. He was a proud member of theGreater Chico Kiwanis Club. He was a backpacker and hiked the entire John MuirTrail in 2001. Frank enjoyed reading, gardening, and playing bridge.
After a losing battle with cancer, Frank passed away Saturday, March 17, 2007,at his home in Chico.
目錄
Foreword
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
1.6 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 ? 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 φ
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 Carath?odory
5.4 Sigma Algebras
5.5 Borel Sets
5.6 Approximating Measurable Sets
5.7 Measurable Functions
5.8 More Measurable Functions
5.9 What Does Monotonicity Tell Uso
5.10 Lebesgue's Differentiation Theorem
5.11 References
6 The Lebesgue 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[-π, π] 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
7.2 Carath?odory'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 Reference
8 The Henstock–Kurzweil Integral
8.1 The Generalized Riemann Integral
8.2 Gauges and δ-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 The 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 Schrodinger Problem A
10.8 An Abstract Cauchy Problem
10.9 Solving in the Schwartz Space
10.10 Solving Schrodinger Problem B
10.11 References
Index
About the Author
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