00. Introduction

Map between Complex and ${\mathbb{R}}^2$

Identify $\mathbb{C}$ with the ${\mathbb{R}}^2$ plane by taking real and imaginary parts so we have

$$z\in \mathbb{C}⟷(x,y)=(\mathrm{Re}z,\mathrm{Im}z)\in {\mathbb{R}}^2$$

Link to original

Distance Metric in $\mathbb{C}$

$$d(z,w)=\sqrt{(\mathrm{Re}z-\mathrm{Re}w{)}^2+(\mathrm{Im}z-\mathrm{Im}w{)}^2}=|z-w|$$

Link to original

Properties of Modulus lemma

Let $z,w\in \mathbb{C}$

1. $|z{|}^2=z\overline{z}$ where $\overline{z}$ is the complex conjugate of $z$
2. If $z=r{e}^{i\theta }$ where $r\in [0,\infty )$ and $\theta \in \mathbb{R}$ then $|z|=r$
3. $|zw|=|z||w|$

Link to original

Open Ball in $\mathbb{C}$

Let $U\subset \mathbb{C}$ be a subset of $\mathbb{C}$ so

$U$ is open if for every $z\in U$ there is some $ϵ>0$ such that open ball

$$B(z,e):=\{w\in C:|z-w|<ϵ\}\subset U$$

so that the ball is contained in $U$

Link to original

Domain

Connected open subset $D\subset \mathbb{C}$ of the complex plane

Link to original