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 $y$:

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

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

Mapping

A function $f$ is aka. 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.
• $\omega = z+i$ translates each point $z$ one unit upwards.
• $\omega = e^{i\theta}$ rotates each point $z$ through an angle $\theta$ about the origin, counterclockwise, since $e^{i\theta} = \cos\theta + i\sin\theta$.
• $\omega = iz$ rotates each non-zero point $z$ through a 90 degree angle about the origin, counterclockwise, since $i = e^{i\pi/2}$.
• $\omega = \bar{z}$ reflects each point $z$ about the real axis.
• $\omega = -z$ reflects each point $z$ about the imaginary axis.

Neighborhood

A neighborhood of $z_0$ is any set containing $D_r(z_0)$ for some $r > 0$.

$D_r(z_0)$ is the open disk of radius $r$ centered at $z_0$.

Deleted Neighborhood

Aka. punctured neighborhood. A deleted neighborhood of $z_0$ is a neighborhood with $z_0$ removed. The deleted disk of radius $r$ is

Open Set and Domain

A set $G \subseteq \mathbb{C}$ is open iff every point of $G$ has a neighborhood contained in $G$.

A domain is an open set that is connected.

Limit Point

Let $D \subseteq \mathbb{C}$ and $z_0 \in \mathbb{C}$. $z_0$ is a limit point of $D$ iff every deleted neighborhood of $z_0$ contains at least 1 point of $D$.

$z_0$ is a limit point of $D$ iff

A limit point of $D$ need not belong to $D$.

Isolated Point

Let $D \subseteq \mathbb{C}$ and $z_0 \in D$. $z_0$ is an isolated point of $D$ iff $z_0$ is not a limit point of $D$. Equivalently, some deleted neighborhood of $z_0$ contains no point of $D$.

$z_0$ is an isolated point of $D$ iff

This is the same as $D_r(z_0) \cap D = \{z_0\}$ for some $r > 0$.

Otherwise the point is a non-isolated point of $D$. It is a limit point of $D$ that belongs to $D$.