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).

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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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. Referenced on Wolfram|Alpha Asymptotic Notation
Cite this as:
Weisstein, Eric W. "Asymptotic Notation."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/AsymptoticNotation.html

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