Landau Symbols
المؤلف:
Bachmann, P
المصدر:
Analytische Zahlentheorie, Bd. 2: Die Analytische Zahlentheorie. Leipzig, Germany: Teubner, 1894. de Bruijn, N. G. Asymptotic Methods in Analysis. New York: Dover
الجزء والصفحة:
...
13-3-2019
1881
Landau Symbols
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 the symbols 
(sometimes called "big-O") and 
(sometimes called "little-o") are known as the Landau symbols and defined as follows.
1. 
means that 
for some constant 
and all values of 
and 
,
2. 
means that 
(Hardy and Wright 1979, pp. 7-8).
Historically speaking, the symbol 
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 
did indeed originate with Landau (1909) in place of the earlier notation ![<span style=]()
{x}" src="http://mathworld.wolfram.com/images/equations/LandauSymbols/Inline18.gif" style="height:14px; width:17px" />(Narkiewicz 2000, p. XI).
REFERENCES:
Bachmann, P. Analytische Zahlentheorie, Bd. 2: Die Analytische Zahlentheorie. Leipzig, Germany: Teubner, 1894.
de 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." http://members.aol.com/jeff570/nth.html.
Landau, 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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