前言
Chapter I Complex Numbers and Functions
1 Complex Numbers
1.1 Complex Number Field
1.2 Complex Plane
1.3 Modulus, Conjugation, Argument, Polar Representation
1.4 Powers and Roots of Complex Numbers
Exercises
2 Regions in the Complex Plane
2.1 Some Basic Concept
2.2 Domain and Jordan Curve
Exercises
3 Functions of a Complex Variable
3.1 The Concept of Functions of a Complex Variable
3.2 Limits and Continuous
Exercises
4 The Extended Complex Plane and the Point at Infinity
4.1 The Spherical Representation, the Extended Complex Plane
4.2 Some Concepts in the Extended Complex Plane
Exercises
Chapter II Analytic Functions
1 The Concept of the Analytic Function
1.1 The Derivative of the Functions of a Complex Variable
1.2 Analytic Functions
Exercises
2 Cauchy-Riemann Equations
Exercises
3 Elementary Functions
3.1 The Exponential Function
3.2 Trigonometric Functions
3.3 Hyperbolic Functions
Exercises
4 Multi-Valued Functions
4.1 The Logarithmic Function
4.2 Complex Power Functions
4. 3 Inverse Trigonometric and Hyperbolic Functions
Exercises
Chapter III Complex Integration
1 The Concept of Contour Integrals
1.1 Integral of a Complex Function over a Real Interval
1.2 Contour Integrals
Exercises
Cauchy-Goursat Theorem
2.1 Cauchy Theorem
2.2 Cauchy Integral Formula
2.3 Derivatives of Analytic Functions
2.4 Liouville''s Theorem and the Fundamental Theorem of Algebra
Exercises
Harmonic Functions
Exercises
Chapter IV Series
1 Basic Properties of Series
1.1 Convergence of Sequences
1.2 Convergence of Series
1.3 Uniform convergence
Exercises
2 Power Series
Exercises
3 Taylor Series
Exercises
4 Laurent Series
Exercises
5 Zeros of an Analytic Functions and Uniquely Determined Analytic
Functions
5.1 Zeros of Analytic Functions
5.2 Uniquely Determined Analytic Functions
5.3 Maximum Modulus Principle
Exercises
6 The Three Types of Isolated Singular Points at a Finite Point
Exercises
7 The Three Types of Isolated Singular Points at a Infinite Point
Exercises
Chapter V Calculus of Residues
1 Residues
1.1 Residues
1.2 Cauchy''s Residue Theorem
1.3 The Calculus of Residue
Exercises
2 Applications of Residue
2.1 The Type of Definite Integral □
2.2 The Type of Improper Integral □
2.3 The Type of Improper Integral □
Exercises
3 Argument Principle
Exercises
Chapter VI Conformal Mappings
1 Analytic Transformation
1.1 Preservation of Domains of Analytic Transformation
1.2 Conformality of Analytic Transformation
Exercises
2 Rational Functions
2.1 Polynomials
2.2 Rational Functions
Exercises
3 Fractional Linear Transformations
Exercises
4 Elementary Conformal Mappings
Exercises
5 The Riemann Mapping Theorem
Exercises
Appendix
Appendix 1
Appendix 2
Answers
Bibliography
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