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Mathematics-Online lexicon:

Landau Symbols O and o

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Let $ f$ and $ g$ be real functions. One writes using the Landau symbol O

$\displaystyle f(x) = O(g(x))\qquad \left(\ x\to a\right)\,, $

if there exists a constant $ c$ such that $ \vert f(x)\vert\le c \vert g(x)\vert$ for $ x$ sufficiently close to $ a$ .

If lim$ _{x \to a} \vert f(x)\vert/\vert g(x)\vert = 0 $ , then the Landau symbol o is used and one writes

$\displaystyle f(x) = o(g(x)) \qquad \left(\ x\to a\right)\, .$

Note that $ a = \pm \infty $ is allowed.

Often the notation $ f(x) \in O(g(x)) $ is used instead of $ f(x) = O(g(x)) $ as well as $ f(x) \in o(g(x)) $ instead of $ f(x) = o(g(x)) $ (always under the assumption $ x \to a $ ).


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see also:

[Examples]
  automatically generated 4/ 7/2008