Complex domain - Knowlpedia

Definition

A complex domain is a nonempty, connected open subset $D\subseteq\mathbb C$ . Connectedness is part of the term “domain”; an arbitrary open subset of $\mathbb C$ may have several components.

Why domains are natural

Local analytic statements can be made on any open set. Global uniqueness results, such as the identity theorem, use connectedness to propagate local information throughout $D$ . Each connected component of an open set is itself a domain, so little is lost by working one component at a time.

Convention

Some authors use “region” as a synonym and reserve “domain” for the source of any function. This knowl uses complex domain only in the topological sense above.

References

John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978. Publisher record. Relevant: Chapter II, §1.