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# Asymptotic Notation

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

1. to mean that for some constant and all values of and ,

2. to mean that ,

3. to mean that ,

4. to mean the same as ,

5. to mean , and

6. to mean for some positive constants and .

implies and is stronger than .

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

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

If , or equivalently or , then and are said to be asymptotically equivalent (Hardy and Wright 1979, p. 8).

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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## References

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. https://mathworld.wolfram.com/AsymptoticNotation.html