Big *O* is the strander mathematical notation. And this notation is denoted by capital O which is tilted at a slight angle on the right side.

In most cases, you will notice the use of *O* symbols instead of in various books or documents.

Symbol | Big O notation |

Type | asymptotic |

Package | mismath, physics |

Commands | `O(arg)` , `\mathcal{O}(arg)` |

Example | `O(log n)` → O(log n) |

```
\documentclass{article}
\begin{document}
$$ O(n^2) $$
$$ O(n^c) $$
$$ O(g(x)) $$
$$ O(\max(g_1,g_2)) $$
\end{document}
```

**Output : **

You have always noticed that backslash is used before the command name. But, in this case, there is no need.

`\mathcal{o}`

will return big o. However, the top-left side of the capital o will remain open in terms of shape.

```
\documentclass{article}
\begin{document}
$$ \mathcal{O}(\log n) $$
$$ \mathcal{O}(\log n^c) $$
$$ \mathcal{O}(n\log n) $$
$$ \mathcal{O}(n!) $$
\end{document}
```

**Output : **

Above two methods do not require any package. However, below we will learn two more ways to represent this symbol with the help of mismath and physics package.

## \bigo and \big0 commands in mismath package

Use of mismath package is unknown to many. But, even then, \bigo and \bigO commands can denote this asymptotic notation.

```
\documentclass{article}
\usepackage{mismath}
\begin{document}
$$ \bigo(m^n) $$
$$ \bigo(1) $$
$$ \bigO(n^3) $$
$$ \bigO(2^n) $$
\end{document}
```

**Output : **

Little o is used as an asymptotic notation like big o. For example

```
\documentclass{article}
\usepackage{mismath}
\begin{document}
$$ f(n)=\lito(g(n)) $$
$$ f\in \lito(g) $$
$$ \lito \le f(n) <c*g(n) $$
\end{document}
```

**Output : **

## Physics package for \order{arg} command

The advantage of \order command is that the size of symbol will increase and decrease according to the size of the argument.

```
\documentclass{article}
\usepackage{physics}
\begin{document}
$$ \order{<n} $$
$$ \order{c^n} $$
$$ \order{n\log n} $$
$$ \order{\order{h(n)}} $$
$$ \order{\frac{n}{k}} $$
\end{document}
```

**Output : **

If you want to manually increase the size of the parenthesis, you can use the \big on the right side of the \order command.

```
\documentclass{article}
\usepackage{physics}
\begin{document}
$$ \order\big{\log_b n} $$
$$ \order\big{\log_c n} $$
$$ \order\big{f(x)} $$
$$ \order\Big{\frac{n}{q}} , \order\bigg{\frac{n_i}{q_k}} , \order\Bigg{\frac{n_{ij}}{q_{kj}}} $$
\end{document}
```

**Output : **

One thing to keep in mind is that the size of the symbol is fixed by using the * sign with each command in the physics package.

```
\documentclass{article}
\usepackage{physics}
\begin{document}
$$ \order*{n^{\frac{5}{2}}} $$
$$ \order*{\frac{n}{k}} $$
$$ \order*{f(x)} $$
\end{document}
```

**Output : **