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.