Asymptotic Notation

Let n be an integer variable which tends to infinity and let x be a continuous variable tending to some limit. Also, let phi(n) or phi(x) be a positive function and f(n) or f(x) any function. Then Hardy and Wright (1979) define

1. f=O(phi) to mean that |f|<Aphi for some constant A and all values of n and x,

2. f=o(phi) to mean that f/phi->0,

3. f∼phi to mean that f/phi->1,

4. f≺phi to mean the same as f=o(phi),

5. f≻phi to mean f/phi->infty, and

6. f=phi to mean A_1phi<f<A_2phi for some positive constants A_1 and A_2.

f=o(phi) implies and is stronger than f=O(phi).

The term Landau symbols is sometimes used to refer the big-O notation O(x) and little-O notation o(x). In general, O(x) and o(x) are read as "is of order x."

If f=phi, then f and phi are said to be of the same order of magnitude (Hardy and Wright 1979, p. 7).

If f∼g, or equivalently f=phi+o(phi) or f=phi(1+o(1)), then f and phi are said to be asymptotically equivalent (Hardy and Wright 1979, p. 8).

See also

Almost All, Asymptotic, Big-O Notation, Big-Omega Notation, Big-Theta Notation, Landau Symbols, Little-O Notation, Order of Magnitude, Tilde

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Hardy, G. H. and Wright, E. M. "Some Notations." §1.6 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 7-8, 1979.Jeffreys, H. and Jeffreys, B. S. "Increasing and Decreasing Functions." §1.065 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 22, 1988.

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Asymptotic Notation

Cite this as:

Weisstein, Eric W. "Asymptotic Notation." From MathWorld--A Wolfram Web Resource.

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