Definition:O Notation/Big-O Notation/Real

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Definition

Estimate at infinity

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


The statement:

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

is equivalent to:

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


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 $g$.


Point Estimate

Let $x_0 \in \R$.

Let $f$ and $g$ be real-valued or complex-valued functions defined on a punctured neighborhood of $x_0$.


The statement:

$f \left({x}\right) = \mathcal O \left({g \left({x}\right)}\right)$ as $x \to x_0$

is equivalent to:

$\exists c \in \R: c \ge 0: \exists \delta \in \R : \delta > 0 : \forall x \in \R : \left({0 < \left\lvert{x - x_0}\right\rvert < \delta \implies \left\lvert{f \left({x}\right)}\right\rvert \le c \cdot \left\lvert{g \left({x}\right)}\right\rvert}\right)$


That is:

$\left\lvert{f \left({x}\right)}\right\rvert \le c \cdot \left\lvert{g \left({x}\right)}\right\rvert$

for all $x$ in a punctured neighborhood of $x_0$.


Also known as

The big-$\OO$ notation, along with little-$\mathcal o$ 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 the functions appearing in the $\OO$-estimate be positive or strictly positive.


Examples

Big-O Estimate for Real Function/Examples

Sources