# 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 ${x}_{0}=+\mathrm{\infty}$. Let $f:D\to \mathbb{R}$ 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

$$\underset{x\to {x}_{0}}{lim}\frac{g(x)}{f(x)}=0.$$ |

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 |