### 1 Complex functions

Let the complex variable $z$ be defined by $z=x+\text{i}y$ where $x$ and $y$ are real variables and $\text{i}$ is, as usual, given by ${\text{i}}^{2}=-1$ . Now let a second complex variable $w$ be defined by $w=u+\text{i}v$ where $u$ and $v$ are real variables. If there is a relationship between $w$ and $z$ such that to each value of $z$ in a given region of the $z-$ plane there is assigned one, and only one, value of $w$ then $w$ is said to be a function of $z$ , defined on the given region. In this case we write

$\phantom{\rule{2em}{0ex}}w=f\left(z\right)$ .

As a example consider $w={z}^{2}-z$ , which is defined for all values of $z$ (that is, the right-hand side can be computed for every value of $z$ ). Then, remembering that $z=x+\text{i}y$ ,

$\phantom{\rule{2em}{0ex}}w=u+\text{i}v={\left(x+\text{i}y\right)}^{2}-\left(x+\text{i}y\right)={x}^{2}+2\text{i}xy-{y}^{2}-x-\text{i}y.$

Hence, equating real and imaginary parts: $\phantom{\rule{1em}{0ex}}u={x}^{2}-x-{y}^{2}$ and $v=2xy-y.$

If $z=2+3\text{i},$ for example, then $x=2,y=3$ so that $u=4-2-9=-7$ and $v=12-3=9$ , giving $w=-7+9\text{i}$ .

##### Example 1
1. For which values of $z$ is $w=\frac{1}{z}$ defined?
2. For these values obtain $u$ and $v$ and evaluate $w$ when $z=2-\text{i}$ .
##### Solution
1. $w$ is defined for all $z\ne 0$ .
2. $\phantom{\rule{1em}{0ex}}u+\text{i}v=\frac{1}{x+\text{i}y}=\frac{1}{x+iy}\cdot \frac{x-\text{i}y}{x-\text{i}y}=\frac{x-\text{i}y}{{x}^{2}+{y}^{2}}.$ Hence $u=\frac{x}{{x}^{2}+{y}^{2}}$ and $v=\frac{-y}{{x}^{2}+{y}^{2}}$ .

If $z=2-\text{i}$ then $x=2,y=-1$ so that ${x}^{2}+{y}^{2}=5$ . Then $u=\frac{2}{5},v=-\frac{1}{5}\text{and}\phantom{\rule{1em}{0ex}}w=\frac{2}{5}-\frac{1}{5}\text{i}$ .