formal definition of Landau notation

Let us consider a domain D and an accumulation pointMathworldPlanetmathPlanetmath x0D¯. Important examples are D= and x0D or D= and x0=+. Let f:D be any functionMathworldPlanetmath. We are going to define the spaces o(f) and O(f) which are families of real functions defined on D and which depend on the point x0D¯.

Suppose first that there exists a neighbourhood U of x0 such that f restricted to UD is always different from zero. We say that go(f) as xx0 if


We say that gO(f) as xx0 if there exists a neighbourhood U of x0 such that

g(x)f(x)is bounded if restricted to DU.

In the case when f0 in a neighbourhood of x0, we define o(f)=O(f) as the set of all functions g which are null in a neighbourhood of 0.

The families o and O are usually called ”small-o” and ”big-o” or, sometimes, ”small ordo”, ”big ordo”.

Title formal definition of Landau notation
Canonical name FormalDefinitionOfLandauNotation
Date of creation 2013-03-22 15:15:48
Last modified on 2013-03-22 15:15:48
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 6
Author paolini (1187)
Entry type Definition
Classification msc 26A12
Synonym Landau notation
Synonym small o
Synonym big o
Synonym order of infinity
Synonym order of zero
Related topic PropertiesOfOAndO