本书以快速和易懂的方式向读者介绍了复变量的主要内容，虽然不能面面俱到，但它确实为读者在这一基础领域打下坚实的基础。书中配有大量的插图和例题，论述生动、引人入胜。本书可作为初学这门学科的本科生或准备参加考试的研究生的重要学习资料。在这部佳作中，Krantz为复变量划了重点。本书有一个包含大约250个名词的极好的术语表和一个供延伸阅读的参考文献。

Preface
1 The Complex Plane
1.1 Complex Arithmetic
1.1.1 The Real Numbers
1.1.2 The Complex Numbers
1.1.3 Complex Conjugate
1.1.4 Modulus of a Complex Number
1.1.5 The Topology of the Complex Plane
1.1.6 The Complex Numbers as a Field
1.1.7 The Fundamental Theorem of Algebra
1.2 The Exponential and Applications
1.2.1 The Exponential Function
1.2.2 The Exponential Using Power Series
1.2.3 Laws of Exponentiation
1.2.4 Polar Form of a Complex Number
1.2.5 Roots of Complex Numbers
1.2.6 The Argument of a Complex Number
1.2.7 Fundamental Inequalities
1.3 Holomorphic Functions
1.3.1 Continuously Differentiable and Ck Functions
1.3.2 The Cauchy-Riemann Equations
1.3.3 Derivatives
1.3.4 Definition of Holomorphic Function
1.3.5 The Complex Derivative
1.3.6 Alternative Terminology for Holomorphic Functions
1.4 Holomorphic and Harmonic Functions
1.4.1 Harmonic Functions
1.4.2 How They are Related
2 Complex Line Integrals
2.1 Real and Complex Line Integrals
2.1.1 Curves
2.1.2 Closed Curves
2.1.3 Differentiable and Ck Curves
2.1.4 Integrals on Curves
2.1.5 The Fundamental Theorem of Calculus along Curves
2.1.6 The Complex Line Integral
2.1.7 Properties of Integrals
2.2 Complex Differentiability and Conformality
2.2.1 Limits
2.2.2 Holomorphicity and the Complex Derivative
2.2.3 Conformality
2.3 The Cauchy Integral Formula and Theorem
2.3.1 The Cauchy Integral Theorem, Basic Form
2.3.2 The Cauchy Integral Formula
2.3.3 More General Forms of the Cauchy Theorems
2.3.4 Deformability of Curves
2.4 The Limitations of the Cauchy Formula
3 Applications of the Cauchy Theory
3.1 The Derivatives of a Holomorphic Function
3.1.1 A Formula for the Derivative
3.1.2 The Cauchy Estimates
3.1.3 Entire Functions and Liouville's Theorem
3.1.4 The Fundamental Theorem of Algebra
3.1.5 Sequences of Holomorphic Functions and their Derivatives
3.1.6 The Power Series Representation of a Holomorphic Function
3.2 The Zeros of a Holomorphic Function
3.2.1 The Zero Set of a Holomorphic Function
3.2.2 Discreteness of the Zeros of a Holomorphic Function
3.2.3 Discrete Sets and Zero Sets
3.2.4 Uniqueness of Analytic Continuation
……
4 Laurent Series
5 The Argument Principle
6 The Geometric Theory
7 Harmonic Functions
8 Infinite Series and Products
9 Analytic Continuation
Glossary of Terms from Complex Variable Theory and Analysis
Bibliography
Index
About the Author