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Importing Wikidata short description: "Mathematical function common in physics" |
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[[Image:Pibmasterplot.png|325px|thumb|'''Figure 1'''. Illustration of a stretched exponential fit (with ''β''=0.52) to an empirical master curve. For comparison, a least squares single and a [[Double exponential function|double exponential]] fit are also shown. The data are rotational [[anisotropy]] of [[anthracene]] in [[polyisobutylene]] of several [[molecular mass]]es. The plots have been made to overlap by dividing time (''t'') by the respective characteristic [[time constant]].]]
The '''stretched exponential function''' <math display="block">f_\beta (t) = e^{ -t^\beta }</math> is obtained by inserting a fractional [[power law]] into the [[exponential function]]. In most applications, it is meaningful only for arguments {{mvar|t}} between 0 and +∞. With {{math|1=''β'' = 1}}, the usual exponential function is recovered. With a ''stretching exponent'' ''β'' between 0 and 1, the graph of log ''f'' versus ''t'' is characteristically ''stretched'', hence the name of the function. The '''compressed exponential function''' (with {{math|1=''β'' > 1}}) has less practical importance, with the notable
In mathematics, the stretched exponential is also known as the [[Cumulative distribution function#Complementary cumulative distribution function (tail distribution)|complementary cumulative]] [[Weibull distribution]]. The stretched exponential is also the [[characteristic function (probability theory)|characteristic function]], basically the [[Fourier transform]], of the [[stable distribution|Lévy symmetric alpha-stable distribution]].
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