Definition:Big-O Notation/Complex/Point

Definition
Let $z_0\in C$.

Let $f$ and $g$ be complex functions defined on an open set containing $z_0$.

The statement:
 * $f(z) = \mathcal O \left({g(z)}\right)$ as $z\to z_0$

is equivalent to the statement:
 * $\displaystyle \exists c\in \R: c\ge 0 : \exists \epsilon\in\R:\epsilon>0 : |f(z)|\leq c\cdot|g(z)|$ for all $z\in N_\epsilon(z_0)$

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