图书信息
图书目录
积分的花园（影印版）
暂无简介
作者：
Frank E. Burk 著
定价：
135.00 元
出版时间：
2026-03-26
ISBN：
978-7-04-065965-8
物料号：
65965-00
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
分析
重点项目：
暂无
版面字数：
460.00千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次印刷时间：
2026-01-09
- 前辅文
- 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 à 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 ø
- 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 Caratheodory
- 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 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 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 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 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
