Definition:Big-O Notation/Complex/Infinity

Definition
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 $\C$.