\begin{definition}[Definition:O Notation/Big-O Notation/Complex/Infinity]
Let $f$ and $g$ be complex functions defined for all complex numbers whose modulus is sufficiently large.
The statement:
:$\map f z = \map \OO {\map g z}$ as $\cmod z \to \infty$
is equivalent to:
:$\exists c \in \R_{\ge 0}: \exists r_0 \in \R: \forall z \in \C: \paren {\cmod z \ge r_0 \implies \cmod {\map f z} \le c \cdot \cmod {\map g z} }$
That is:
:$\cmod {\map f z} \le c \cdot \cmod {\map g z}$
for all $z$ in a neighborhood of infinity in $\C$.
Category:Definitions/Complex Analysis
Category:Definitions/Asymptotic Notation
Category:Definitions/Order Notation
\end{definition}