# Definition:O Notation

## Definition

$\OO$ notation is a type of order notation, typically used in computer science for comparing 'run-times' of algorithms, or in analysis for comparing growth rates between two growth functions.

### Big-O Notation

Let $f$ and $g$ be real-valued or complex-valued functions defined on a neighborhood of $+ \infty$ in $\R$.

The statement:

$\map f x = \map \OO {\map g x}$ as $x \to \infty$

is equivalent to:

$\exists c \in \R_{\ge 0}: \exists x_0 \in \R: \forall x \in \R: \paren {x \ge x_0 \implies \size {\map f x} \le c \cdot \size {\map g x} }$

That is:

$\size {\map f x} \le c \cdot \size {\map g x}$

for $x$ sufficiently large.

This statement is voiced $f$ is big-O of $g$ or simply $f$ is big-O $g$.

### Little-O Notation

Let $\map g x \ne 0$ for $x$ sufficiently large.

$f$ is little-o of $g$ as $x \to \infty$ if and only if:

$\ds \lim_{x \mathop \to \infty} \ \frac {\map f x} {\map g x} = 0$

## Also known as

The $\OO$ and $\mathcal o$ used in $\OO$ notation can be referred to as the Landau symbols.