Definition:Connected Domain (Complex Analysis)

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Let $D \subseteq \C$ be a subset of the set of complex numbers.

Then $D$ is a connected domain if and only if $D$ is open and connected.

Simply Connected Domain

Let $D \subseteq \C$ be a connected domain.

Then $D$ is called a simply connected domain if and only if $D$ is simply connected.

Also defined as

A connected domain $D$ is often used as the domain of a complex-differentiable function $f: D \to \C$.

Also known as

Some texts omit the word connected and simply call $D$ a domain.

Also see