Definition:Big-O Notation/Complex

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Definition

Estimate at 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 $\C$.


Point Estimate

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 $z_0$.


Also known as

The big-$\OO$ notation, along with little-$\oo$ notation, are also referred to as Landau's symbols or the Landau symbols, for Edmund Georg Hermann Landau.


In analytic number theory, sometimes Vinogradov's notations $f \ll g$ or $g \gg f$ are used to mean $f = \map \OO g$.

This can often be clearer for estimates leading to typographically complex error terms.


Some sources use an ordinary $O$:

$f = \map O g$


Also defined as

Some authors require that, in the expression $f = \map \OO g$, the function $g$ be real-valued and positive or strictly positive.