Complex Analysis (Second Edition)

Author: Lu Jianke
Publishing House: Wuhan University Press
Brief Introduction:
This book is based on the outline for the compilation of "Complex Analysis (with an emphasis on applications)" drafted by the editing group of the Ministry of Education's former Committee on the Compilation and Review of Science and Engineering Textbooks in Mathematics and Mechanics between 1987 and 1989. The book consists of nine chapters, covering complex numbers and complex functions, fundamentals of analytic functions, integration, series, residues, analytic continuation, conformal mapping, harmonic functions, and applications of analytic functions.
As an attempt, this book includes new content of practical value such as higher-order singular integrals and extended residue theorems. It presents a fresh approach to multivalued functions, introduces new proofs and descriptions for theorems such as Cauchy's theorem (in homotopy form), the argument principle, conformal mapping, and the uniqueness theorem for analytic functions. Modernized treatments or improvements of traditional content are integrated throughout the chapters. Practical teaching experience over the years has demonstrated that it is a highly teachable and learnable textbook.
This book is suitable for undergraduate students in foundational mathematics, applied mathematics, computational mathematics, mechanics, astronomy, and mathematics education at comprehensive universities, as well as graduate students in certain engineering disciplines. It can also serve as a reference for physics majors, engineering professionals, and self-learners.
Table of Contents:
Chapter 1: Complex Numbers and Complex Functions
1.1 Complex Numbers
1.1.1 The Field of Complex Numbers
1.1.2 Geometric Representation of Complex Numbers
1.1.3 Stereographic Projection, Riemann Sphere, Infinity, Extended Complex Plane
Exercises 1.1
1.2 Complex Functions
1.2.1 Concept of Complex Functions
1.2.2 Limits and Continuity of Complex Functions
1.2.3 Homotopy and Connectivity of Regions
1.2.4 Argument Function
Exercises 1.2
1.3 Complex Sequences and Series
1.3.1 Complex Sequences and Series with Complex Terms
1.3.2 Complex Function Sequences and Series
Exercises 1.3
Exercises for Chapter 1
Chapter 2: Fundamentals of Analytic Functions
2.1 Analytic Functions
2.1.1 Derivative and Its Geometric Meaning
2.1.2 Analytic Functions
Exercises 2.1
2.2 Some Elementary Analytic Functions
2.2.1 Polynomials and Rational Functions
2.2.2 Exponential Functions
2.2.3 Trigonometric and Hyperbolic Functions
2.2.4 Logarithmic Functions
2.2.5 Power Functions and Radical Functions
2.2.6 Branch Points of Elementary Multivalued Functions
2.2.7 Logarithms of Rational Functions
2.2.8 Roots of Rational Functions
2.2.9 Inverse Trigonometric and Hyperbolic Functions
Exercises 2.2
Exercises for Chapter 2
Chapter 3: Complex Integration
3.1 Concept of Complex Integration
3.1.1 Definition and Computation of Complex Integration
3.1.2 Basic Properties of Complex Integration
Exercises 3.1
3.2 Fundamental Theorems
3.2.1 Cauchy's Integral Theorem
3.2.2 Antiderivatives
Exercises 3.2
3.3 Fundamental Formulas
3.3.1 Cauchy's Integral Formula
3.3.2 Cauchy's Derivative Formula
3.3.3 Cauchy's Inequality
3.3.4 Morera's Theorem
Exercises 3.3
3.4 Improper Complex Integrals
3.4.1 Definition of Improper Complex Integrals
3.4.2 Cauchy Principal Value Integrals
3.4.3 Higher-Order Singular Integrals
Exercises 3.4
Exercises for Chapter 3
Chapter 4: Series Theory of Analytic Functions
4.1 General Theory
Chapter 5: Residue Theory
Chapter 6: Analytic Continuation
Chapter 7: Conformal Mapping
Chapter 8: Harmonic Functions
Chapter 9: Applications of Analytic Functions in Plane Fields
Appendix 1: Proof of Sufficiency for the Determination of Single-Valued Branches of Elementary Multivalued Functions
Appendix 2: Detailed Explanation of the Origin of the Definition of Higher (Integer) Order Singular Integrals
Solutions or Hints for Exercises