📂Complex Anaylsis

Representation of Complex Numbers in Polar Coordinates

Definition ¹

A complex number $z \ne 0$ corresponds to a point $P(x,y)$ on the complex plane, and can be expressed in polar coordinates through the length $r := |z|$ of segment $\overline{OP}$ and the counterclockwise angle $\theta$ made by the segment $\overline{OP}$ and the $x$ axis. $$z = r \left( \cos \theta + i \sin \theta \right)$$ In this context, $\theta$ is called the Argument and is represented as $\theta = \arg z$. A complex number corresponds to infinitely many arguments $\theta + 2n \pi$, but by convention, the unique argument that satisfies $\pi < \theta \le \pi$ is called the Principal Argument and is represented as $\theta = \arg z$.

Description

• Depending on the literature, $z = r \left( \cos \theta + i \sin \theta \right)$ is sometimes abbreviated as $z = r \text{cis} \theta$.
• The original term for Argument in computing is often used to mean a parameter, among other things. While localizing it to “편각” is not inherently bad as it emphasizes the geometric aspect, it is advisable to stick to the English pronunciation, especially when studying from English materials.