formal definition of Landau notation

Let us consider a domain $D$ and an accumulation point $x_{0}\in\overline{D}$. Important examples are $D=\mathbb{R}$ and $x_{0}\in D$ or $D=\mathbb{N}$ and $x_{0}=+\infty$. Let $f\colon D\to\mathbb{R}$ be any function. 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 $x_{0}\in\overline{D}$.

Suppose first that there exists a neighbourhood $U$ of $x_{0}$ such that $f$ restricted to $U\cap D$ is always different from zero. We say that $g\in o(f)$ as $x\to x_{0}$ if

 $\lim_{x\to x_{0}}\frac{g(x)}{f(x)}=0.$

We say that $g\in O(f)$ as $x\to x_{0}$ if there exists a neighbourhood $U$ of $x_{0}$ such that

 $\frac{g(x)}{f(x)}\text{is bounded if restricted to D\cap U}.$

In the case when $f\equiv 0$ in a neighbourhood of $x_{0}$, 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