Complex Analysis — Introduction
Fundamentals of Complex Numbers and Functions
High School – 1st Year UniversityLearning Objectives
- Understand the definition and arithmetic of complex numbers
- Master the complex plane and polar form
- Understand the meaning and applications of Euler's formula
- Learn the basic properties of complex functions
- Become familiar with complex exponential, trigonometric, and logarithmic functions
Prerequisites
- High school mathematics (trigonometry, exponentials and logarithms)
- Basic calculus (the concept of limits)
Chapter Overview
Chapter 1: Definition and Arithmetic of Complex Numbers
Learn the imaginary unit $i$, the four arithmetic operations on complex numbers, conjugates, and absolute values.
Chapter 2: The Complex Plane and Polar Form
Learn representation on the complex (Gauss) plane, polar form $r(\cos\theta + i\sin\theta)$, and de Moivre's theorem.
Chapter 3: Euler's Formula
Learn Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ and Euler's identity $e^{i\pi} + 1 = 0$.
Chapter 4: Basics of Complex Functions
Learn what complex functions are, their real and imaginary parts, and the concepts of limits and continuity.
Chapter 5: Complex Exponential and Trigonometric Functions
Learn the complex exponential $e^z$, complex trigonometric functions $\sin z$, $\cos z$, and their relation to hyperbolic functions.
Chapter 6: The Complex Logarithm
Learn the definition of the complex logarithm, multi-valuedness and principal value, branches and branch cuts, and the power function $z^\alpha$.
Chapter 7: Exercises
A collection of exercises to test your understanding of the introductory material.
Overview
Complex analysis is the calculus of functions whose variables are complex numbers. Extending real analysis to the complex domain reveals a surprisingly beautiful theory.
Complex numbers were introduced to solve equations that have no real solutions, such as $x^2 + 1 = 0$. Using the imaginary unit $i = \sqrt{-1}$, the solutions are expressed as $x = \pm i$.
One of the most beautiful results of complex analysis is Euler's formula:
$$e^{i\theta} = \cos\theta + i\sin\theta$$This formula connects the exponential function with trigonometric functions, which at first glance appear to be unrelated. Setting $\theta = \pi$ yields the famous Euler's identity:
$$e^{i\pi} + 1 = 0$$This identity unites five of the most important constants in mathematics ($e$, $i$, $\pi$, $1$, $0$) in a single equation and is often called "the most beautiful equation in mathematics."
This introduction covers the basic properties of complex numbers through the fundamentals of complex functions. This knowledge forms the foundation for studying more advanced topics in complex analysis, such as Cauchy's integral theorem and the residue theorem.