Lecture Notes on Complex Analysis

The following sections are included:

• Preface

• Contents

The following sections are included:

• Informal Introduction

• Complex Plane

• Properties of the Modulus

• The Argument of a Complex Number

• Formal Construction of Complex Numbers

• The Riemann Sphere and the Extended Complex Plane

The following sections are included:

• Complex Sequences

• Subsequences

• Convergence of Sequences

• Cauchy Sequences

• Complex Series

• Absolute Convergence

• nth-Root Test

• Ratio Test

The following sections are included:

• Open Discs and Interior Points

• Closed Sets

• Limit Points

• Closure of a Set

• Boundary of a Set

• Cantor's Theorem

• Compact Sets

• Polygons and Paths in ℂ

• Connectedness

• Domains

The following sections are included:

• Complex-Valued Functions

• Continuous Functions

• Complex Differentiable Functions

• Cauchy-Riemann Equations

• Analytic Functions

• Power Series

• The Derived Series

• Identity Theorem for Power Series

The following sections are included:

• The Functions exp z, sin z and cos z

• Complex Hyperbolic Functions

• Properties of exp z

• Properties of sin z and cos z

• Addition Formulae

• The Appearance of π

• Inverse Trigonometric Functions

• More on exp z and the Zeros of sin z and cos z

• The Argument Revisited

• Arg z is Continuous in the Cut-Plane

The following sections are included:

• Introduction

• The Complex Logarithm and its Properties

• Complex Powers

• Branches of the Logarithm

The following sections are included:

• Paths and Contours

• The Length of a Contour

• Integration along a Contour

• Basic Estimate

• Fundamental Theorem of Calculus

• Primitives

The following sections are included:

• Cauchy's Theorem for a Triangle

• Cauchy's Theorem for Star-Domains

• Deformation Lemma

• Cauchy's Integral Formula

• Taylor Series Expansion

• Cauchy's Integral Formulae for Derivatives

• Morera's Theorem

• Cauchy's Inequality and Liouville's Theorem

• Identity Theorem

• Preservation of Angles

The following sections are included:

• Laurent Expansion

• Uniqueness of the Laurent Expansion

The following sections are included:

• Isolated Singularities

• Behaviour near an Isolated Singularity

• Behaviour as |z| → ∞

• Casorati-Weierstrass Theorem

The following sections are included:

• Residues

• Winding Number (Index)

• Cauchy's Residue Theorem

The following sections are included:

• Zeros and Poles

• Argument Principle

• Rouché's Theorem

• Open Mapping Theorem

The following sections are included:

• Mean Value Property

• Maximum Modulus Principle

• Minimum Modulus Principle

• Functions on the Unit Disc

• Hadamard's Theorem and the Three Lines Lemma

The following sections are included:

• Special Transformations

• Inversion

• Möbius Transformations

• Möbius Transformations in the Extended Complex Plane

The following sections are included:

• Harmonic Functions

• Local Existence of a Harmonic Conjugate

• Maximum and Minimum Principle

The following sections are included:

• Local Uniform Convergence

• Hurwitz's Theorem

• Vitali's Theorem

The following sections are included:

• Appendix A Some Results from Real Analysis

• Bibliography

• Index