In real analysis, a branch of mathematics, a slowly varying function is a function of a real variable whose behaviour at infinity is in some sense similar to the behaviour of a function converging at infinity. Similarly, a regularly varying function is a function of a real variable whose behaviour at infinity is similar to the behaviour of a power law function (like a polynomial) near infinity. These classes of functions were both introduced by Jovan Karamata,[1] [2] and have found several important applications, for example in probability theory.
. A measurable function is called slowly varying (at infinity) if for all,
\limx
| L(ax) | |
| L(x) |
=1.
. Let . Then is a regularly varying function if and only if
\foralla>0,gL(a)=\limx
| L(ax) | |
| L(x) |
\inR+
These definitions are due to Jovan Karamata.[1]
Regularly varying functions have some important properties:[1] a partial list of them is reported below. More extensive analyses of the properties characterizing regular variation are presented in the monograph by .
. The limit in and is uniform if is restricted to a compact interval.
. Every regularly varying function is of the form
f(x)=x\betaL(x)
Note. This implies that the function in has necessarily to be of the following form
g(a)=a\rho
. A function is slowly varying if and only if there exists such that for all the function can be written in the form
L(x)=\exp\left(η(x)+
| x | |
| \int | |
| B |
| \varepsilon(t) | |
| t |
dt\right)
where
\limxL(x)=b\in(0,infty),
then is a slowly varying function.