Definition:Big-O Notation/Complex/Point

Definition

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 $z_0$.