# Asymptotic formula

From Encyclopedia of Mathematics

A formula containing the symbols $o$, $O$ or the equivalence sign $\sim$ (asymptotic equality of functions).

Examples of asymptotic formulas:

$$\sin x=x+o(x^2),\quad x\to0;$$

$$\cos x=1+O(x^2),\quad x\to0;$$

$$x^3+x+1\sim x^3,\quad x\to\infty;$$

$$\pi(x)\sim\frac{x}{\ln x},\quad x\to\infty$$

($\pi(x)$ is the amount of prime numbers not exceeding $x$).

#### Comments

On the meaning of the symbols $o, O$ and $\sim$, see e.g. [a1] or [a2].

#### References

[a1] | N.G. de Bruijn, "Asymptotic methods in analysis" , Dover, reprint (1981) |

[a2] | A. Erdélyi, "Asymptotic expansions" , Dover, reprint (1956) |

**How to Cite This Entry:**

Asymptotic formula.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_formula&oldid=32389

This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article