\begin{definition}[Definition:O Notation/Big-O Notation/Complex/Point]
Let $z_0 \in \C$.
Let $f$ and $g$ be complex functions defined on a punctured neighborhood of $z_0$.
The statement:
:$\map f z = \map \OO {\map g z}$ as $z \to z_0$
is equivalent to:
:$\exists c \in \R_{\ge 0}: \exists \delta \in \R_{>0}: \forall z \in \C : \paren {0 < \cmod {z - z_0} < \delta \implies \cmod {\map f z} \le c \cdot \cmod {\map g z} }$
That is:
:$\cmod {\map f z} \le c \cdot \cmod {\map g z}$
for all $z$ in a punctured neighborhood of $z_0$.
Category:Definitions/Asymptotic Notation
Category:Definitions/Order Notation
Category:Definitions/Complex Analysis
\end{definition}