LECTURE NOTES OF WILLIAM CHEN
INTRODUCTION TO COMPLEX ANALYSIS
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.
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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