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).
