Asymptotic formula
From Encyclopedia of Mathematics
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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 $x$).
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
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