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 the statement:
 * $\displaystyle \exists c\in \R: c\ge 0 : \exists r_0\in\R : |f(z)|\leq c\cdot|g(z)|$ for all $z\in\C$ with $|z|\geq r_0$