Domain (of the plane)

An open, connected set -- the standard setting for defining a complex function.

A domain is an open, connected subset of the complex plane. This pairing of properties is exactly what most theorems in this course need to hold: openness guarantees every point has room to take a derivative or a limit in every direction, and connectedness guarantees the domain is one piece, so a property established near one point can be carried, step by step, to any other point of it. Almost every complex function in this course is defined on a domain.

Examples

The open disk $|z|<1$ is a domain.
The punctured plane $\mathbb{C}\setminus\{0\}$ (all $z\ne 0$ ) is a domain: it is open, and still connected since a path can route around the missing origin.