§3.1. The Exponential Function §3.2. The Logarithmic Function §3.3. Branches and Derivatives of Logarithms §3.4. Some Identities on Logarithms §3.5. Complex Power Functions §3.6. Trigonometric Functions §3.7. Hyperbolic Functions §3.8. Inverse Trigonometric and Hyperbolic Functions

§5.1 孤立奇点 1. 孤立奇点定义 2. 孤立奇点分类 3. 孤立奇点性质 4. 零点与极点的关系 §5.2 留数  1. 留数的定义和定理  2. 留数的计算规则  3. 在无穷远点的留数 §5.3 留数在定积分计算的应用 §5.4 对数留数与幅角原理 1、对数留数 2、辐角原理 3、路西定理 4、小结与思考

§6.1 共形映射的概念  1. 曲线的切线  2. 解析函数的导数的几何意义  3. 共形映射的概念 §6.2 分式线性映射  1. 分式线性映射的定义  2. 分式线性映射的性质 §6.3 唯一决定分式线性映射的条件 §6.4 几个初等函数构成的映射  1. 幂函数  2. 指数函数

§5.1. Convergence of Series §5.2. Taylor Series §5.3. Examples §5.4. Laurent Series §5.5. Examples §5.6. Absolute and Uniform Convergence of Power Series §5.7. Continuity of Sums of Power Series §5.8. Integration and Differentiation of Power Series §5.9. Uniqueness of Series Representations §5.10. Multiplication and Division of Power Series

§6.1. Residues §6.2. Cauchy’s Residue Theorem §6.3. Using a Single Residue §6.4. The Three Types of Isolated Singular Points §6.5. Residues at Poles §6.6. Examples §6.7. Zeros of Analytic Functions §6.8. Uniquely Determined Analytic Functions §6.9. Zeros and Poles §6.10. Behavior of f Near Isolated Singular Points §6.11. Reflection Principle

§7.1. Evaluation of Improper Integrals §7.2. Examples §7.3. Improper Integrals From Fourier Analysis §7.4. Jordan’s Lemma §7.5. Indented Paths §7.6. An Indentation Around a Branch Point §7.7. Definite Integrals Involving Sine and Cosine §7.8. Argument Principle §7.9. Rouche’s Theorem

§8.1. Conformal mappings §8.2. Unilateral Functions §8.3. Local Inverses §8.4. Affine Transformations §8.5. The Transformation = /1 zw §8.6. Mappings by /1 z §8.7. Fractional Linear Transformations §8.8. Cross Ratios §8.9. Mappings of the Upper Half Plane

§4.1. Derivatives of Complex-Valued Functions of §4.2. Definite Integrals of Functions w §4.3. Paths §4.4. Path Integrals §4.5. Examples §4.6. Upper Bounds for Integrals §4.7. Primitive Functions §4.8. Examples §4.9. Cauchy Integral Theorem §4.10. Proof of Cauchy Integral Theorem §4.11. Extended Cauchy Integral Theorem §4.12. Cauchy Integral Formula §4.13. Derivatives of Analytic Functions §4.14. Liouville’s Theorem §4.15. Maximum Modulus Principle

一、发展史: 复变函数理论被人誉为19世纪最独特 的创造,这个新的数学分支统治了19世纪。几 乎象微积分的直接扩展统治了18世纪那样,曾 被称为19世纪的数学享受,也曾被称为抽象科 学最和谐的理论之一

一、指数函数 二、对数函数 三、乘幂 ab与幂函数 四、三角函数和双曲函数 五、反三角函数和反双曲函数 六、小结与思考