# Slowly varying function > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Slowly_varying_function > Markdown URL: https://mediated.wiki/source/Slowly_varying_function.md > Source: https://en.wikipedia.org/wiki/Slowly_varying_function > Source revision: 1339846850 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Function in mathematics}} In [real analysis](/source/real_analysis), a branch of [mathematics](/source/mathematics), a '''slowly varying function''' is a [function of a real variable](/source/function_of_a_real_variable) whose behaviour at [infinity](/source/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](/source/power_law) function (like a [polynomial](/source/polynomial)) near infinity. These classes of functions were both introduced by [Jovan Karamata](/source/Jovan_Karamata),See {{harv|Galambos|Seneta|1973}}See {{harv|Bingham|Goldie|Teugels|1987}}. and have found several important applications, for example in [probability theory](/source/probability_theory) and [extreme value theory](/source/extreme_value_theory). == Basic definitions == {{EquationRef|1|Definition 1}}. A [measurable function](/source/measurable_function) {{math|''L'' : (0, +∞) → (0, +∞)}} is called ''slowly varying'' (at infinity) if for all {{math|''a'' > 0}}, :\lim_{x \to \infty} \frac{L(ax)}{L(x)}=1. {{EquationRef|2|Definition 2}}. Let {{math|''L'' : (0, +∞) → (0, +∞)}}. Then {{math|''L''}} is a regularly varying function if and only if \forall a > 0, g_L(a) = \lim_{x \to \infty} \frac{L(ax)}{L(x)} \in \mathbb{R}^{+}. In particular, the [limit](/source/limit_of_a_function) must be finite. These definitions are due to [Jovan Karamata](/source/Jovan_Karamata). == Basic properties == Regularly varying functions have some important properties: 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 1}} and {{EquationNote|2|2}} is [uniform](/source/uniform_convergence) if {{mvar|''a''}} is restricted to a compact [interval](/source/Interval_(mathematics)). ===Karamata's characterization theorem=== {{EquationRef|4|Theorem 2}}. Every regularly varying function {{math|''f'' : (0, +∞) → (0, +∞)}} is of the form :f(x)=x^\beta L(x) where *{{mvar|β}} is a [real number](/source/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 :g(a)=a^\rho where the real number {{mvar|''ρ''}} 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 :L(x) = \exp \left( \eta(x) + \int_B^x \frac{\varepsilon(t)}{t} \,dt \right) where *{{math|''η''(''x'')}} is a [bounded](/source/Bounded_function) [measurable function](/source/measurable_function) of a real variable converging to a finite number as {{mvar|''x''}} goes to infinity *{{math|''ε''(''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 ::\lim_{x \to \infty} L(x) = b \in (0,\infty), :then {{mvar|''L''}} is a slowly varying function. * For any {{math|''β'' ∈ '''R'''}}, the function {{math|''L''(''x'') {{=}} log{{hairsp}}''β''{{hairsp}}''x''}} is slowly varying. * The function {{math|''L''(''x'') {{=}} ''x''}} is not slowly varying, nor is {{math|''L''(''x'') {{=}} ''x''{{hairsp}}''β''}} for any real {{math|''β ''≠ 0}}. However, these functions are regularly varying. ==See also== *[Analytic number theory](/source/Analytic_number_theory) *[Hardy–Littlewood tauberian theorem](/source/Hardy%E2%80%93Littlewood_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](/source/Cambridge) | publisher=[Cambridge University Press](/source/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](/source/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 --- Adapted from the Wikipedia article [Slowly varying function](https://en.wikipedia.org/wiki/Slowly_varying_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Slowly_varying_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.