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{{Short description|Function in mathematics}}
In [[real analysis]], a branch of [[mathematics]], a '''slowly varying function''' is a [[function of a real variable]] whose behaviour at [[Infinity#Real analysis|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]],<ref name="GaSe" >See {{harv|Galambos|Seneta|1973}}</ref><ref name="BiGoTe">See {{harv|Bingham|Goldie|Teugels|1987}}.</ref> and have found several important applications, for example in [[probability theory]].

== Basic definitions ==

{{EquationRef|1|Definition 1}}. A [[measurable function]] {{math|''L''&nbsp;:&nbsp;(0, +&infin;)&nbsp;&rarr;&nbsp;(0, +&infin;)}} is called ''slowly varying'' (at infinity) if for all {{math|''a'' > 0}},
:<math>\lim_{x \to \infty} \frac{L(ax)}{L(x)}=1.</math>

{{EquationRef|2|Definition 2}}. Let {{math|''L''&nbsp;:&nbsp;(0, +&infin;)&nbsp;&rarr;&nbsp;(0, +&infin;)}}. Then {{math|''L''}} is a regularly varying function if and only if <math>\forall a > 0, g_L(a) = \lim_{x \to \infty} \frac{L(ax)}{L(x)} \in \mathbb{R}^{+}</math>. In particular, the [[limit of a function|limit]] must be finite.

These definitions are due to [[Jovan Karamata]].<ref name="GaSe" /><ref name="BiGoTe" />

== Basic properties ==

Regularly varying functions have some important properties:<ref name="GaSe" /> a partial list of them is reported below. More extensive analyses of the properties characterizing regular variation are presented in the monograph by {{harvtxt|Bingham|Goldie|Teugels|1987}}.

===Uniformity of the limiting behaviour===
{{EquationRef|3|Theorem 1}}. The limit in {{EquationNote|1|definitions&nbsp;1}} and {{EquationNote|2|2}} is [[uniform convergence|uniform]] if {{mvar|''a''}} is restricted to a compact [[Interval (mathematics)|interval]].

===Karamata's characterization theorem===
{{EquationRef|4|Theorem 2}}. Every regularly varying function {{math|''f''&nbsp;:&nbsp;(0, +&infin;)&nbsp;&rarr;&nbsp;(0, +&infin;)}} is of the form 
:<math>f(x)=x^\beta L(x)</math> 
where 
*{{mvar|β}} is a [[real number]],
*{{mvar|L}} is a slowly varying function. 
'''Note'''. This implies that the function {{math|''g''(''a'')}} in {{EquationNote|2|definition 2}} has necessarily to be of the following form 
:<math>g(a)=a^\rho</math>
where the real number {{mvar|''&rho;''}} is called the ''index of regular variation''.

===Karamata representation theorem===
{{EquationRef|5|Theorem 3}}. A function {{mvar|''L''}} is slowly varying if and only if there exists {{math|''B'' > 0}} such that for all {{math|''x'' ≥ ''B''}} the function can be written in the form

:<math>L(x) = \exp \left( \eta(x) + \int_B^x \frac{\varepsilon(t)}{t} \,dt \right)</math>

where 
*{{math|''&eta;''(''x'')}} is a [[Bounded function|bounded]] [[measurable function]] of a real variable converging to a finite number as {{mvar|''x''}} goes to infinity
*{{math|''&epsilon;''(''x'')}} is a bounded measurable function of a real variable converging to zero as {{mvar|''x''}} goes to infinity.

== Examples ==
* If {{mvar|''L''}} is a measurable function and has a limit
::<math>\lim_{x \to \infty} L(x) = b \in (0,\infty),</math> 
:then {{mvar|''L''}} is a slowly varying function.
* For any {{math|''&beta;'' &isin; '''R'''}}, the function {{math|''L''(''x'')&nbsp;{{=}}&nbsp;log<sup>&hairsp;''&beta;''</sup>&hairsp;''x''}} is slowly varying.
* The function {{math|''L''(''x'')&nbsp;{{=}}&nbsp;''x''}} is not slowly varying, nor is {{math|''L''(''x'')&nbsp;{{=}}&nbsp;''x''<sup>&hairsp;''&beta;''</sup>}} for any real {{math|''&beta;&nbsp;''≠&nbsp;0}}.  However, these functions are regularly varying.

==See also==
*[[Analytic number theory]]
*[[Hardy–Littlewood tauberian theorem]] and its treatment by Karamata

==Notes==
{{reflist|30em}}

==References==
* {{SpringerEOM|title=Karamata theory|oldid=25937|first=N.H.|last=Bingham}}
* {{Citation
| last=Bingham
| first=N. H.
| last2=Goldie
| first2=C. M.
| last3=Teugels
| first3=J. L.
| title=Regular Variation
| place=[[Cambridge]]
| publisher=[[Cambridge University Press]]
| series=Encyclopedia of Mathematics and its Applications
| volume=27
| year=1987
| edition=
| url=https://archive.org/details/regularvariation0000bing
| doi=
| isbn=0-521-30787-2
| mr=0898871
| zbl=0617.26001
| url-access=registration
}}
* {{Citation |author2-link=Eugene Seneta | last1=Galambos | first1=J. | last2=Seneta | first2=E. | title=Regularly Varying Sequences | year=1973 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=41 | issue=1 | pages=110–116 | doi=10.2307/2038824 | jstor=2038824| doi-access=free }}.

[[Category:Real analysis]]
[[Category:Tauberian theorems]]
[[Category:Types of functions]]