Definition:Big-O Notation/Complex/Point

Definition
Let $z_0 \in \C$.

Let $f$ and $g$ be complex functions defined on an punctured neighborhood of $z_0$.

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

is equivalent to the statement:
 * $\displaystyle \exists c \in \R : c \ge 0 : \exists \delta \in \R : \delta > 0 : \forall z \in \C : (0<|z-z_0|<\delta \implies |f(z)| \leq c \cdot |g(z)|)$

That is, $|f(z)| \leq c \cdot |g(z)|$ for all $z$ in a punctured neighborhood of $z_0$.