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 and little-o notations  and , and in general, and  are read as "is of order ."

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.