LECTURE NOTES OF WILLIAM CHEN
# INTRODUCTION TO COMPLEX ANALYSIS

### Chapter 1 : COMPLEX NUMBERS >>

### Chapter 2 : FOUNDATIONS OF COMPLEX ANALYSIS >>

### Chapter 3 : COMPLEX DIFFERENTIATION >>

### Chapter 4 : COMPLEX INTEGRALS >>

### Chapter 5 : CAUCHY'S INTEGRAL THEOREM >>

### Chapter 6 : CAUCHY'S INTEGRAL FORMULA >>

### Chapter 7 : TAYLOR SERIES, UNIQUENESS AND THE MAXIMUM PRINCIPLE >>

### Chapter 8 : ISOLATED SINGULARITIES AND LAURENT SERIES >>

### Chapter 9 : CAUCHY'S INTEGRAL THEOREM REVISITED >>

### Chapter 10 : RESIDUE THEORY >>

### Chapter 11 : EVALUATION OF DEFINITE INTEGRALS >>

### Chapter 12 : HARMONIC FUNCTIONS AND CONFORMAL MAPPINGS >>

### Chapter 13 : MÖBIUS TRANSFORMATIONS >>

### Chapter 14 : SCHWARZ-CHRISTOFFEL TRANSFORMATIONS >>

### Chapter 15 : LAPLACE'S EQUATION REVISITED >>

### Chapter 16 : UNIFORM CONVERGENCE >>

This set of notes has been organized in such a way to create a single volume suitable for an introduction to some of the basic ideas in complex analysis. The material in Chapters 1 - 11 and 16 were used in various forms between 1981 and 1990 by the author at Imperial College, University of London. Chapters 12 - 15 were added in Sydney in 1996.

To read the notes, click the links below for connection to the appropriate PDF files.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

- Arithmetic and Conjugates
- Polar Coordinates
- Rational Powers

- Three Approaches
- Point Sets in the Complex Plane
- Complex Functions
- Extended Complex Plane
- Limits and Continuity

- Introduction
- The Cauchy-Riemann Equations
- Analytic Functions
- Introduction to Special Functions
- Periodicity and its Consequences
- Laplace's Equation and Harmonic Conjugates

- Curves in the Complex Plane
- Contour Integrals
- Inequalities for Contour Integrals
- Equivalent Curves
- Riemann Sums

- A Restricted Case
- Analytic Functions in a Star Domain
- Nested Triangles
- Further Examples

- Introduction
- Derivatives
- Further Consequences

- Remarks on Series
- Taylor Series
- Uniqueness
- The Maximum Principle

- Removable Singularities
- Poles
- Essential Singularities
- Isolated Singularities at Infinity
- Further Examples
- Laurent Series

- Simply Connected Domains
- Cauchy's Integral Theorem
- Cauchy's Integral Formula
- Analytic Logarithm

- Cauchy's Residue Theorem
- Finding the Residue
- Principle of the Argument

- Introduction
- Rational Functions over the Unit Circle
- Rational Functions over the Real Line
- Rational and Trigonometric Functions over the Real Line
- Bending Round a Singularity
- Integrands with Branch Points

- A Local Property of Analytic Functions
- Laplace's Equation
- Global Properties of Analytic Functions

- Linear Functions
- The Inversion Function
- A Generalization
- Finding Particular Möbius Transformations
- Symmetry and Möbius Transformations

- Introduction
- A Generalization
- Polygons
- Examples

- Use of Möbius Transformations
- Use of Schwarz-Christoffel Transformations

- Uniform Convergence of Sequences
- Consequences of Uniform Convergence
- Cauchy Sequences
- Uniform Convergence of Series
- Application to Power Series
- Cauchy Sequences