古典分析导引（影印版）

本书对古典分析中的特定主题做了严格处理，提供了许多应用和示例。书中内容基于高等微积分的基本原理，不涉及复分析和Lebesgue积分等更复杂的技术，适合本科水平阅读；涵盖的主题包括：Fourier级数和积分、近似理论、Fourier公式、Γ函数、Bernoulli数和多项式、Riemann zeta函数、Tauber定理、椭圆积分、Cantor集的分支，以及微分方程的理论探讨，包括正则奇点的幂级数解、Bessel函数、超几何函数和Sturm比较理论。预备章节快速回顾了基本原理和更多的背景知识，例如无穷乘积和常用不等式。本书适合读者自学，但也可用作高等微积分课程第二学期的教材。每章末尾配有大量的练习题。历史注记讨论了数学思想的演变及其在物理中的应用。本书的特色在于穿插了重要数学家的人物小传和画像。尽管本书为本科生编写，但其他读者可以收获关于经典主题的素材，这些主题是纯粹数学和应用数学近代发展的基础。

Preface
Chapter 1. Basic Principles
1.1. Mathematical induction
1.2. Real numbers
1.3. Completeness principles
1.4. Numerical sequences
1.5. Infinite series
1.6. Continuous functions and derivatives
1.7. The Riemann integral
1.8. Uniform convergence
1.9. Historical remarks
1.10. Metric spaces
1.11. Complex numbers
Exercises
Chapter 2. Special Sequences
2.1. The number e
2.2. Irrationality of m
2.3. Euler's constant
2.4. Vieta's product formula
2.5. Wallis product formula
2.6. Stirling's formula
Exercises
Chapter 3. Power Series and Related Topics
3.1. General properties of power series
3.2. Abel's theorem
3.3. Cauchy products and Mertens'theorem
3.4. Taylor's formula with remainder
3.5. Newton's binomial series
3.6. Composition of power series
3.7. Euler's sum
3.8. Continuous nowhere differentiable functions
Exercises
Chapter 4. Inequalities
4.1. Elementary inequalities
4.2. Cauchy's inequality
4.3. Arithmetic-geometric mean inequality
4.4. Integral analogues
4.5. Jensen's inequality
4.6. Hilbert's inequality
Exercises
Chapter 5. Infinite Products
5.1. Basic concepts
5.2. Absolute convergence
5.3. Logarithmic series
5.4. Uniform convergence
Exercises
Chapter 6. Approximation by Polynomials
6.1. Interpolation
6.2. Weierstrass approximation theorem
6.3. Landau's proof
6.4. Bernstein polynomials
6.5. Best approximation
6.6. Stone–Weierstrass theorem
6.7. Refinements of Weierstrass theorem
Exercises
Chapter 7. Tauberian Theorems
7.1. Summation of divergent series
7.2. Tauber's theorem
7.3. Theorems of Hardy and Littlewood
7.4. Karamata's proof
7.5. Hardy's power series
Exercises
Chapter 8. Fourier Series
8.1. Physical origins
8.2. Orthogonality relations
8.3. Mean-square approximation
8.4. Convergence of Fourier series
8.5. Examples
8.6. Gibbs' phenomenon
8.7. Arithmetic means of partial sums
8.8. Continuous functions with divergent Fourier series
8.9. Fourier transforms
8.10. Inversion of Fourier transforms
8.11. Poisson summation formula
Exercises
Chapter 9. The Gamma Function
9.1. Probability integral
9.2. Gamma function
9.3. Beta function
9.4. Legendre's duplication formula
9.5. Euler's reflection formula
9.6. Infinite product representation
9.7. Generalization of Stirling's formula
9.8. Bohr-Mollerup theorem
9.9. A special integral
Exercises
Chapter 10. Two Topics in Number Theory
10.1. Equidistributed sequences
10.2. Weyl's criterion
10.3. The Riemann zeta function
10.4. Connection with the gamma function
10.5. Functional equation
Exercises
Chapter 11. Bernoulli Numbers
11.1. Calculation of Bernoulli numbers
11.2. Sums of positive powers
11.3. Euler's sums
11.4. Bernoulli polynomials
11.5. Euler-Maclaurin summation formula
11.6. Applications of Euler-Maclaurin formula
Exercises
Chapter 12. The Cantor Set
12.1. Cardinal numbers
12.2. Lebesgue measure
12.3. The Cantor set
12.4. The Cantor-Scheeffer function
12.5. Space-filling curves
Exercises
Chapter 13. Differential Equations
13.1. Existence and uniqueness of solutions
13.2. Wronskians
13.3. Power series solutions
13.4. Bessel functions
13.5. Hypergeometric functions
13.6. Oscillation and comparison theorems
13.7. Refinements of Sturm's theory
Exercises
Chapter 14. Elliptic Integrals
14.1. Standard forms
14.2. Fagnano's duplication formula
14.3. The arithmetic-geometric mean
14.4. The Legendre relation
Exercises
Index of Names
Subject Index