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