Asymptotic formula

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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 $x$).


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


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