Definition:Deleted Neighborhood/Complex Analysis
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Definition
Let $z_0 \in \C$ be a point in the complex plane.
Let $\map {N_\epsilon} {z_0}$ be the $\epsilon$-neighborhood of $z_0$.
Then the deleted $\epsilon$-neighborhood of $z_0$ is defined as $\map {N_\epsilon} {z_0} \setminus \set {z_0}$.
That is, it is the $\epsilon$-neighborhood of $z_0$ with $z_0$ itself removed.
It can also be defined as:
- $\map {N_\epsilon} {z_0} \setminus \set {z_0} : = \set {z \in A: 0 < \cmod {z_0 - z} < \epsilon}$
from the definition of $\epsilon$-neighborhood.
Also known as
A deleted neighborhood is also called a punctured neighborhood.
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $1.$