Definition:Deleted Neighborhood/Complex Analysis

Definition
Let $z_0 \in \C$ be a point in the complex plane.

Let $N_\epsilon \left({z_0}\right)$ be the $\epsilon$-neighborhood of $z_0$.

Then the deleted $\epsilon$-neighborhood of $z_0$ is defined as $N_\epsilon \left({z_0}\right) \setminus \left\{{z_0}\right\}$.

That is, it is the $\epsilon$-neighborhood of $z_0$ with $z_0$ itself removed.

It can also be defined as:
 * $N_\epsilon \left({z_0}\right) \setminus \left\{{z_0}\right\} : = \left\{{z \in A: 0 < \left \vert{z_0 - z}\right \vert < \epsilon}\right\}$

from the definition of $\epsilon$-neighborhood.