复变量导引（影印版）

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