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.
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.
Chapter 1: COMPLEX
NUMBERS
- Arithmetic and Conjugates
- Polar Coordinates
- Rational Powers
Chapter 2: FOUNDATIONS
OF COMPLEX ANALYSIS
- Three Approaches
- Point Sets in the Complex Plane
- Complex Functions
- Extended Complex Plane
- Limits and Continuity
Chapter 3: COMPLEX
DIFFERENTIATION
- 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
- Curves in the Complex Plane
- Contour Integrals
- Inequalities for Contour Integrals
- Equivalent Curves
- Riemann Sums
Chapter 5: CAUCHY'S
INTEGRAL THEOREM
- A Restricted Case
- Analytic Functions in a Star Domain
- Nested Triangles
- Further Examples
Chapter 6: CAUCHY'S
INTEGRAL FORMULA
- Introduction
- Derivatives
- Further Consequences
Chapter 7: TAYLOR
SERIES, UNIQUENESS AND THE MAXIMUM PRINCIPLE
- Remarks on Series
- Taylor Series
- Uniqueness
- The Maximum Principle
Chapter 8: ISOLATED
SINGULARITIES AND LAURENT SERIES
- Removable Singularities
- Poles
- Essential Singularities
- Isolated Singularities at Infinity
- Further Examples
- Laurent Series
Chapter 9: CAUCHY'S
INTEGRAL THEOREM REVISITED
- Simply Connected Domains
- Cauchy's Integral Theorem
- Cauchy's Integral Formula
- Analytic Logarithm
Chapter 10: RESIDUE
THEORY
- Cauchy's Residue Theorem
- Finding the Residue
- Principle of the Argument
Chapter 11: EVALUATION
OF DEFINITE INTEGRALS
- 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
- A Local Property of Analytic Functions
- Laplace's Equation
- Global Properties of Analytic Functions
Chapter 13: MÖBIUS
TRANSFORMATIONS
- Linear Functions
- The Inversion Function
- A Generalization
- Finding Particular Möbius Transformations
- Symmetry and Möbius Transformations
Chapter 14: SCHWARZ-CHRISTOFFEL
TRANSFORMATIONS
- Introduction
- A Generalization
- Polygons
- Examples
Chapter 15: LAPLACE'S
EQUATION REVISITED
- Use of Möbius Transformations
- Use of Schwarz-Christoffel Transformations
Chapter 16: UNIFORM
CONVERGENCE
- Uniform Convergence of Sequences
- Consequences of Uniform Convergence
- Cauchy Sequences
- Uniform Convergence of Series
- Application to Power Series
- Cauchy Sequences