Complex Analysis

Introduction

Complex analysis is a branch of mathematics that extends the concepts of calculus to complex numbers. Instead of dealing with real numbers, we work with numbers of the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit (the square root of $-1$). Complex analysis has numerous applications in physics, engineering, number theory, and applied mathematics. It is particularly useful in solving problems related to fluid dynamics, electromagnetism, and quantum mechanics.

Contents

Complex Differentiation and Conformality

Complex Numbers
Fundamental Theorem of Algebra
Complex Differentiability
Holomorphic and Conformal Maps
The Riemann Sphere
Möbius Transformations

Complex Integration

Integration and Primitives
The Homotopy Invariance Theorem
Cauchy Integral Formula for Disks
Cauchy Integral Formula for Jordan Domains

Complex Series and Series Expansion

Normal Convergence of Holomorphisms
Series of Complex Functions
Series Expansion of Holomorphic Functions
Laurent Series
Isolated Singularities
Residue

The Riemann Mapping Theorem

The Argument Principle
The Open Mapping Theorem
Conformal Transformations of The Disk
Riemann Mapping Theorem

Acknowledgements

This page follows the structure of the course MATH3228 at ANU in 2024.

References and Useful Resources:

Alexander Isaev, Twenty-One Lectures on Complex Analysis
José Figueroa-O’Farrill, Complex Analysis
Samuel J. Li, Complex Function Plotter
Juan Carlos Ponce Campuzano, Complex Analysis