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

$w=f\left(z\right)$ .

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 $z=x+\text{i}y$ ,

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

Hence, equating real and imaginary parts: $u=x^2-x-y^2$ 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. $u+\text{i}v=\frac{1}{x+\text{i}y}=\frac{1}{x+iy}\cdot \frac{x-\text{i}y}{x-\text{i}y}=\frac{x-\text{i}y}{x^2+y^2}.$ 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 $u=\frac{2}{5},v=-\frac{1}{5}\text{and}w=\frac{2}{5}-\frac{1}{5}\text{i}$ .