Landau Symbols

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 the symbols O(x) (sometimes called "big-O") and o(x) (sometimes called "little-o") are known as the Landau symbols and defined as follows.

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

2. f=o(phi) means that f/phi->0

(Hardy and Wright 1979, pp. 7-8).

Historically speaking, the symbol O(x) first appeared in the second volume of Bachmann's treatise on number theory (Bachmann 1894), and Landau obtained this notation in Bachmann's book (Landau 1909, p. 883; Derbyshire 2004, p. 238). However, the symbol o(x) did indeed originate with Landau (1909) in place of the earlier notation {x} (Narkiewicz 2000, p. XI).

See also

Asymptotic Notation, Big-Omega Notation, Big-Theta Notation

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Bachmann, P. Analytische Zahlentheorie, Bd. 2: Die Analytische Zahlentheorie. Leipzig, Germany: Teubner, Bruijn, N. G. Asymptotic Methods in Analysis. New York: Dover, pp. 3-10, 1981.Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.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.Havil, J. "Big Oh Notation." Appendix B in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 219, 2003.Miller, J. "Earliest Uses of Symbols of Number Theory.", E. Handbuch der Lehre von der Verteilung der Primzahlen. Leipzig, Germany: Teubner, 1909. Reprinted by New York: Chelsea, 1953.Narkiewicz, W. The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. New York: Springer-Verlag, 2000.

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Landau Symbols

Cite this as:

Weisstein, Eric W. "Landau Symbols." From MathWorld--A Wolfram Web Resource.

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