# Complex Analysis

## Complex Numbers

### Definition - Rectangular Form

$$z=a+bi$$ where $$a,b\in\mathbb{R}$$ and $$i$$ satisfies $$i^2=-1$$.

## Complex Conjugate

### Definition

$$\overline{z} = a-bi$$.

## Basic Algebra

• Addition. $$z_1+z_2 = (a_1+a_2) + (b_1+b_2)i$$
• Multiplication. $$z_1z_2 = (a_1a_2 - b_1b_2) + (a_1b_2 + a_2b_1)i$$
• Division. $$\frac{z_1}{z_2} = \frac{z_1\overline{z}_2}{z_2\overline{z}_2} = \frac{z_1\overline{z}_2}{a_2^2+b_2^2} = \frac{(a_1a_2+b_1b_2) + (a_2b_1-a_1b_2)i}{a_2^2+b_2^2}$$

## Exponential Representation - Polar Form

### Definition

$$z=|z|\exp(i\theta)$$, $$\theta = \arctan(b/a)$$, $$|z| = \sqrt{z\overline{z}}$$

### Conjugate

$$\overline{z} = |z|\exp(-i\theta)$$

## Argand Diagram

`Complex Plane’

\textbf{Re}al and Complex (\textbf{Im}) axes.

Thus, complex numbers can be represented as a vector in the complex plane.

## Geometry

$$+(z_1,z_2) = (a_1+b_1) + (a_2+b_2)i$$ - vector tail-tip addition

$$|z_1 + z_2| \leq |z_1| + |z_2|$$

### Subtraction - Tail-Tip addition using the opposite direction of the second operand

$$-(z_1,z_2) = +(z_1,-z_2) = (a_1-b_1) + (a_2-b_2)i$$

### Subtraction Length Relation

$$|z_1 - z_2| \geq |z_1| - |z_2|$$

### Multiplication

$$z_1z_2 = r_1r_2\exp(i(\theta_1+\theta_2))$$

## Functions of a Complex Variable

Created: 2024-05-30 Thu 21:15

Validate