comparison of functions, - relations, asymptotic relations
A notion arising in studies on the behaviour of a function with respect to another function in a neighbourhood of some point (this point may be infinite).
Let be a limit point of a set . If for two functions and there exist constants and such that for , , then is called a function which is bounded in comparison with in some deleted neighbourhood of , and this is written as
(read "f is of the order of g" ); means that the considered property holds only in some deleted neighbourhood of . This definition can be naturally used when , .
If two functions and are such that and as , then they are called functions of the same order as . For instance, if two functions are such that , if and if the limit
exists, then they are of the same order as .
Two functions and are called equivalent (asymptotically equal) as (written as ) if in some neighbourhood of , except maybe the point itself, a function is defined such that
The condition of equivalency of two functions is symmetric, i.e. if , then as , and transitive, i.e. if and , then as . If in some neighbourhood of the point the inequalities , hold for , then (*) is equivalent to any of the following conditions:
If where , then is said to be an infinitely-small function with respect to , and one writes
(read "a is of lower order than f" ). If when , then if . If is an infinitely-small function for , one says that the function is an infinitely-small function of higher order than as . If and are quantities of the same order, then one says that is a quantity of order with respect to . All formulas of the above type are called asymptotic estimates; they are especially interesting for infinitely-small and infinitely-large functions.
Examples: (); ; (; any positive numbers); ().
Here are some properties of the symbols and :
if and , then
Formulas containing the symbols and are read only from the left to the right; however, this does not exclude that certain formulas remain true when read from the right to the left. The symbols and for functions of a complex variable and for functions of several variables are introduced in the same way as it was done above for functions of one real variable.
The symbols and ( "little oh symbollittle oh" and "big Oh symbolbig Oh" ) were introduced by E. Landau.
|[a1]||G.H. Hardy, "A course of pure mathematics" , Cambridge Univ. Press (1975)|
|[a2]||E. Landau, "Grundlagen der Analysis" , Akad. Verlagsgesellschaft (1930)|
Order relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Order_relation&oldid=44949