# Complex Analysis Topics

## 1. The Set of Complex Numbers

##### 1.1 Basic Properties of Complex Numbers

- The Set of Complex Numbers
- Addition and Multiplication of Complex Numbers ( Examples 1 )
- Division of Complex Numbers ( Examples 1 )
- The Set of Complex Numbers as an Algebraic Field ( Examples 1 )
- Conjugation of Complex Numbers
- Square Roots of Complex Numbers
- The Absolute Value of Complex Numbers
- Basic Properties of Complex Numbers Review

##### 1.2. The Polar Representation of Complex Numbers

- The Polar Representation of Complex Numbers ( Examples 1 )
- Polar Representation with Multiplication of Complex Numbers
- De Moivre's Formula for the Polar Representation of Powers of Complex Numbers ( Examples 1 )
- nth Roots of Complex Numbers ( Examples 1 )
- nth Roots of Unity
- Polar Representation of Complex Number Review

## 2. Topological Properties of the Set of Complex Numbers

##### 2.1. Sequences of Complex Numbers

- Sequences of Complex Numbers
- Properties of Convergent Sequences of Complex Numbers
- Cauchy Sequences of Complex Numbers

##### 2.2. Open and Closed Sets in the Complex Plane

- Open Sets in the Complex Plane
- Closed Sets in the Complex Plane
- The Interior of a Set of Complex Numbers
- The Closure of a Set of Complex Numbers

##### 2.3. Limits and Continuity of Complex Functions

## 3. Elementary Complex Functions

##### 3.1. The Complex Exponential Function

- The Complex Exponential Function ( Examples 1 )
- Properties of the Complex Exponential Function
- The Graph of the Complex Exponential Function

##### 3.2. The Complex Cosine and Sine Functions

- The Complex Cosine and Sine Functions ( Examples 1 )
- Properties of the Complex Cosine and Sine Functions

##### 3.3. The Complex Natural Logarithm Function

- The Complex Natural Logarithm Function ( Examples 1 )
- Properties of the Complex Natural Logarithm Function

##### 3.4. Complex Power Functions

## 4. Differentiability and Analyticity of Complex Functions

##### 4.1. Differentiable and Analytic Complex Functions

- Differentiable Complex Functions
- Analytic Complex Functions ( Examples 1 )
- Analyticity of Sums, Differences, Products, and Quotients of Analytic Functions
- Analyticity of Polynomial and Rational Complex Functions
- Analyticity of Compositions of Analytic Functions
- The Cauchy-Riemann Theorem ( Examples 1 | Examples 2 )
- Harmonic Functions
- Harmonicity of the Real and Imaginary Parts of an Analytic Complex Function ( Examples 1 )
- Harmonic Conjugates of Analytic Complex Functions

## 5. Curve Integrals of Complex Functions and Cauchy's Theorem

##### 5.1. The Fundamental Theorem of Calculus

- Piecewise Smooth Curves in the Complex Plane
- Integrals of Complex Functions Along Piecewise Smooth Curves ( Examples 1 | Examples 2 )
- Properties of Integrals of Complex Functions Along Piecewise Smooth Curves
- The Arc Length of a Piecewise Smooth Curve
- The Fundamental Theorem of Calculus for Integrals of Complex Functions Along Piecewise Smooth Curves

##### 5.2. Cauchy's Integral Theorems

## 6. Complex Series

##### 6.1. Series of Complex Numbers and Complex Functions

- Sequences and Series of Complex Numbers
- Sequences and Series of Complex Functions
- Convergence Tests for Series of Positive Real Numbers
- Complex Power Series
- The Radius of Convergence for Complex Power Series

##### 6.2. Taylor Series of Analytic Complex Functions

- Taylor Series of Analytic Complex Functions
- Taylor's Theorem for Analytic Complex Functions
- Table of Common Complex Taylor Series

- Orders of Roots of Analytic Complex Functions
- The Coincidence Principle
- Singularities of Analytic Complex Functions
- Criterion for Classifying Isolated Removable Singularities of Analytic Complex Functions
- Criterion for Classifying Isolated Pole Singularities of Analytic Complex Functions

##### 6.3. Laurent Series of Analytic Complex Functions

- Laurent Series of Analytic Complex Functions
- Laurent's Theorem for Analytic Complex Functions
- The Residue of an Analytic Function at a Point ( Examples 1 )
- The Residue of an Analytic Function at a Pole Singularity
- The Residue Theorem
- Evaluating Definite Integrals of Type 1
- Evaluating Definite Integrals of Type 2
- Evaluating Definite Integrals of Type 3
- The Principle of the Argument
- Rouche's Theorem