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:
 * $f(z) = \mathcal O \left({g(z)}\right)$ as $|z|\to\infty$

is equivalent to:
 * $\displaystyle \exists c\in \R: c\ge 0 : \exists r_0 \in \R : \forall z \in \C : (|z| \geq r_0 \implies |f(z)| \leq c \cdot |g(z)|)$

That is:
 * $|f(z)| \leq c \cdot |g(z)|$

for all $z$ in a neighborhood of infinity in $\CC$.