Definition:Neighborhood (Complex Analysis)

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This page is about Neighborhood in the context of Complex Analysis. For other uses, see Neighborhood.

It has been suggested that this page be renamed. In particular: Epsilon Neighborhood (Complex Analysis) To discuss this page in more detail, feel free to use the talk page.

Definition

Let $z_0 \in \C$ be a complex number.

Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.

The $\epsilon$-neighborhood of $z_0$ is defined as:

$\map {N_\epsilon} {z_0} := \set {z \in \C: \cmod {z - z_0} < \epsilon}$

Also known as

A neighborhood in this context is often referred to as an open disk (UK spelling: open disc).

Some sources introduce this concept as $\delta$-neighborhood (that is: delta), but it is the same thing.

Also see

• Complex Plane is Metric Space: this definition is compatible with that of an open $\epsilon$-ball neighborhood in a metric space.

Linguistic Note

The UK English spelling of neighborhood is neighbourhood.

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Point Sets: Complex Numbers: $1.$