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 (Narkiewicz 2000, p. XI).