Introduction to Complex Analysis

by WWL Chen

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 chapters below for connection to the appropriate PDF files. You will need Adobe Acrobat Reader Version 4.0 or later.

The material is available free to all individuals, on the understanding that it is not to be used for financial gains, 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.

Chapter 1: COMPLEX NUMBERS (last uploaded on 2 December 2002)

• Arithmetic and Conjugates
• Polar Coordinates
• Rational Powers

Chapter 2: FOUNDATIONS OF COMPLEX ANALYSIS (last uploaded on 2 December 2002)

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

Chapter 3: COMPLEX DIFFERENTIATION (last uploaded on 2 December 2002)

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

Chapter 4: COMPLEX INTEGRALS (last uploaded on 2 December 2002)

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

Chapter 5: CAUCHY'S INTEGRAL THEOREM (last uploaded on 2 December 2002)

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

Chapter 6: CAUCHY'S INTEGRAL FORMULA (last uploaded on 2 December 2002)

• Introduction
• Derivatives
• Further Consequences

Chapter 7: TAYLOR SERIES, UNIQUENESS AND THE MAXIMUM PRINCIPLE (last uploaded on 2 December 2002)

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

Chapter 8: ISOLATED SINGULARITIES AND LAURENT SERIES (last uploaded on 2 December 2002)

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

Chapter 9: CAUCHY'S INTEGRAL THEOREM REVISITED (last uploaded on 2 December 2002)

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

Chapter 10: RESIDUE THEORY (last uploaded on 2 December 2002)

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

Chapter 11: EVALUATION OF DEFINITE INTEGRALS (last uploaded on 2 December 2002)

• 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

Chapter 12: HARMONIC FUNCTIONS AND CONFORMAL MAPPINGS (last uploaded on 2 December 2002)

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

Chapter 13: MÖBIUS TRANSFORMATIONS (last uploaded on 2 December 2002)

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

Chapter 14: SCHWARZ-CHRISTOFFEL TRANSFORMATIONS (last uploaded on 2 December 2002)

• Introduction
• A Generalization
• Polygons
• Examples

Chapter 15: LAPLACE'S EQUATION REVISITED (last uploaded on 2 December 2002)

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

Chapter 16: UNIFORM CONVERGENCE (last uploaded on 2 December 2002)

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