本书是在“高等微积分”的水平上阐述数学分析中的论题，提供了从初等微积分向实变函数论及复变函数论中的高等课程的一种过渡，而且介绍了某些涉及现代分析的抽象理论。内容既涵盖我国大学的数学分析课程的内容，又包括勒贝格积分及柯西定理和留数计算等。本书条理清晰，内容精练，言简意赅，适合作为高等院校本科生数学分析课程的教材。

汤姆·M.阿波斯托尔（Tom M.Apostol），是加州理工学院数学系荣誉教授。他于1946年在华盛顿大学西雅图分校获得数学硕士学位，于1948年在加州大学伯克利分校获得数学博士学位。

Chapter 1 The Real and Complex Number Systems
1.1 Introduction
1.2 The field axioms
1.3 The order axioms
1.4 Geometric representation of real numbers
1.5 Intervals
1.6 Integers
1.7 The unique factorization theorem for integers
1.8 Rational numbers
1.9 Irrational numbers
1.10 Upper bounds, maximum element, least upper bound (supremum)
1.11 The completeness axiom
1.12 Some properties of the supremum
1.13 Properties of the integers deduced from the completeness axiom
1.14 The Archimedean property of the real-number system
1.15 Rational numbers with finite decimal representation
1.16 Finite decimal approximations to real numbers
1.17 Infinite decimal representation of real numbers
1.18 Absolute values and the triangle inequality
1.19 The Cauchy-Schwarz inequality
1.20 Plus and minus infinity and the extended real number system R*
1.21 Complex numbers
1.22 Geometric representation of complex numbers
1.23 The imaginary unit
1.24 Absolute value of a complex number
1.25 Impossibility of ordering the complex numbers
1.26 Complex exponentials
1.27 Further properties of complex exponentials
1.28 The argument of a complex number
1.29 Integral powers and roots of complex numbers
1.30 Complex logarithms
1.31 Complex powers
1.32 Complex sines and cosines
1.33 Infinity and the extended complex plane C*
Exercises
Chapter 2 Some Basic Notions of Set Theory
2.1 Introduction
2.2 Notations
2.3 Ordered pairs
2.4 Cartesian product of two sets
2.5 Relations and functions
2.6 Further terminology concerning functions
2.7 One-to-one functions and inverses
2.8 Composite functions
2.9 Sequences
2.10 Similar (equinumerous) sets
2.11 Finite and infinite sets
2.12 Countable and uncountable sets
2.13 Uncountability of the real-number system
2.14 Set algebra
2.15 Countable collections of countable sets
Exercises
Chapter 3 Elements of Point Set Topology
3.1 Introduction
3.2 Euclidean space Rn
3.3 Open balls and open sets in Rn
3.4 The structure of open sets in Rt
3.5 Closed sets
3.6 Adherent points. Accumulation points
3.7 Closed sets and adherent points
3.8 The Bolzano-Weierstrass theorem
3.9 The Cantor intersection theorem
3.10 The Lindelofcovering theorem
3.11 The Heine-Borel covering theorem
3.12 Compactness in Rn
3.13 Metric spaces
3.14 Point set topology in metric spaces
3.15 Compact subsets of a metric space
3.16 Boundary of a set
Exercises
Chapter 4 Limits and Continuity
4.1 Introduction
4.2 Convergent sequences in a metric space
4.3 Cauchy sequences
4.4 Complete metric spaces
4.5 Limit of a function
4.6 Limits of complex-valued functions
Chapter 5 Derivatives
Chapter 6 Functions of Bounded Variation and Rectifiable Curves
Chapter 7 The Riemann-Stieltjes Integral
Chapter 8 Infinite Series and Infinite Products
Chapter 9 Sequences of Functions
Chapter 10 The Lebesgue Integral
Chapter 11 Fourier Series and Fourier Integrals
Chapter 12 Multivariable Differential Calculus
Chapter 13 Implicit Functions and Extremum Problems
Chapter 14 Multiple Riemann Integrals
Chapter 15 Multiple Lebesgue Integrals
Chapter 16 Cauchy's Theorem and the Residue Calculus
Index of Special Symbols
Index