Introduction to Complex Functions

Let $S$ be a set of complex numbers. A function $f: S \to \mathbb{C}$ is a rule that assigns to each $z \in S$ a complex number $\omega$ , written $\omega = f(z)$ . $S$ is the domain of $f$ .

Real and Imaginary Parts

If $z = x+iy$ , $f(z)$ decomposes into a pair of real-valued functions of $x$ and $y$ :

f(z) = u(x,y) + iv(x,y)

$u$ is the real part of $f$ , $v$ is the imaginary part of $f$ .

If $z = re^{i\theta}$ ,

f(z) = f(re^{i\theta}) = u(r,\theta) + iv(r,\theta)

Mapping

A function $f$ is also called a mapping or transformation. The image of a point $z \in S$ is $\omega = f(z)$ .

If $T \subseteq S$ , the image of $T$ is the set of images of all points in $T$ .
The image of the entire domain $S$ is the range of $f$ .
The inverse image of a point $\omega$ is the set of all $z$ in the domain of $f$ with $f(z) = \omega$ .

Examples:

$\omega = z+1$ translates each point $z$ one unit to the right.
$\omega = iz$ rotates each non-zero point $z$ through a right angle about the origin, counterclockwise, since $i = e^{i\pi/2}$ .
$\omega = \bar{z}$ reflects each point $z$ in the real axis.