积分的花园（影印版）

样章

作者：

Frank E. Burk 著

定价：

135.00 元

版面字数：

460.00千字

开本：

16开

装帧形式：

精装

页数：

296

最新
印次时间：

2026年3月

ISBN：

978-7-04-065965-8

物料号：

65965-00

出版时间：

2026年3月

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

暂无

目录
- 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