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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
版次：
1
最新版次
印刷时间：
2026-01-09
ISBN：
978-7-04-065965-8
物料号：
65965-00
出版时间：
2026-03-26
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
  • 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

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