# Order relation

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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.