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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 [[Laplace distribution|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.]]
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| The '''stretched exponential function'''
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| :<math>f_\beta (t) = e^{ -t^\beta }</math>
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| is obtained by inserting a fractional [[power law]] into the [[exponential function]].
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| In most applications, it is meaningful only for arguments ''t'' between 0 and +∞. With ''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'', whence the name of the function. The '''compressed exponential function''' (with ''β''>1) has less practical importance, with the notable exception of β=2, which gives the [[normal distribution]].
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| In mathematics, the stretched exponential is also known as the [[Cumulative_distribution_function#Complementary_cumulative_distribution_function_(tail_distribution)|complementary cumulative]] [[Weibull distribution]]. Furthermore, the stretched exponential is the [[characteristic function (probability theory)|characteristic function]] (basically the [[Fourier transform]]) of the [[stable distribution|Lévy symmetric alpha-stable distribution]].
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| In physics, the stretched exponential function is often used as a phenomenological description of [[Relaxation (physics)|relaxation]] in disordered systems. It was first introduced by [[Rudolf Kohlrausch]] in 1854 to describe the discharge of a capacitor;<ref>{{cite journal
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| | author = Kohlrausch, R.
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| | year = 1854
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| | title = Theorie des elektrischen Rückstandes in der Leidner Flasche
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| | journal = [[Annalen der Physik und Chemie|Annalen der Physik und Chemie (Poggendorff)]]
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| | volume = 91
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| | pages = 56–82, 179–213
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| | url = http://gallica.bnf.fr/ark:/12148/bpt6k15176w.pagination}}.</ref>
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| therefore it is also called the '''Kohlrausch function'''. In 1970, G. Williams and D.C. Watts used the [[Fourier transform]] of the stretched exponential to describe [[dielectric spectroscopy|dielectric spectra]] of polymers;<ref>{{cite journal
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| | author = Williams, G. and Watts, D. C.
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| | year = 1970
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| | title = Non-Symmetrical Dielectric Relaxation Behavior Arising from a Simple Empirical Decay Function
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| | journal = Transactions of the [[Faraday Society]]
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| | volume = 66
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| | pages = 80–85
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| | doi = 10.1039/tf9706600080
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| }}.</ref>
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| in this context, the stretched exponential or its Fourier transform are also called the '''Kohlrausch-Williams-Watts (KWW) function'''.
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| In phenomenological applications, it is often not clear whether the stretched exponential function should apply to the differential or to the integral distribution function -- or to neither.
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| In each case one gets the same asymptotic decay, but a different power law prefactor, which makes fits more ambiguous than for simple exponentials. In a few cases <ref>{{cite journal
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| | author = Donsker, M. D. and Varadhan, S. R. S.
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| | journal = Comm. Pure Appl. Math.
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| | volume = 28
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| | pages = 1–47
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| | year = 1975
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| | title = Asymptotic evaluation of certain Markov process expectations for large time
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| }}</ref> <ref>{{cite journal
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| | author = Takano, H. and Nakanishi, H. and Miyashita, S.
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| | journal = Phys. Rev. B
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| | volume = 37
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| | pages = 3716 - 3719
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| | year = 1988
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| | title = Stretched exponential decay of the spin-correlation function in the kinetic Ising model below the critical temperature
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| |bibcode = 1988PhRvB..37.3716T |doi = 10.1103/PhysRevB.37.3716 }}</ref>
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| it can be shown that the asymptotic decay is a stretched exponential, but the prefactor is usually an unrelated power.
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| == Mathematical properties ==
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| === Moments ===
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| Following the usual physical interpretation, we interpret the function argument ''t'' as a time, and ''f''<sub>β</sub>(''t'') is the differential distribution. The area under the curve
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| is therefore interpreted as a ''mean relaxation time''. One finds
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| :<math>\langle\tau\rangle \equiv \int_0^\infty dt\, e^{ - \left( {t /\tau_K } \right)^\beta } = {\tau_K \over \beta }\Gamma ({1 \over \beta })</math>
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| where Γ is the [[gamma function]]. For exponential decay, 〈τ〉 = τ<sub>''K''</sub> is recovered.
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| The higher [[moment (mathematics)|moments]] of the stretched exponential function are:<ref>I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. ''Table of Integrals, Series, and Products'', fourth edition. Academic Press, 1980. Integral 3.478.</ref>
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| : <math>\langle\tau^n\rangle \equiv \int_0^\infty dt\, t^{n-1}\, e^{ - \left( {t /\tau_K } \right)^\beta } = {{\tau_K }^n \over \beta }\Gamma ({n \over \beta })</math>.
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| ===Distribution function===
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| In physics, attempts have been made to explain stretched exponential behaviour as a linear superposition of simple exponential decays. This requires a nontrivial distribution of relaxation times, ''ρ(u)'', which is implicitly defined by
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| : <math>e^{ - t^\beta} = \int_0^\infty du\,\rho(u)\, e^{-t/u}</math>.
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| Alternatively, a distribution
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| : <math>G=u \rho (u)\,</math>
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| is used.
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| ρ can be computed from the series expansion:<ref>{{cite journal
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| | author = Lindsey, C. P. and Patterson, G. D.
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| | year = 1980
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| | title = Detailed comparison of the Williams-Watts and Cole-Davidson functions
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| | journal = [[Journal of Chemical Physics]]
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| | volume = 73
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| | pages = 3348–3357
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| | doi = 10.1063/1.440530|bibcode = 1980JChPh..73.3348L }}.
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| For a more recent and general discussion, see {{cite journal
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| | author = Berberan-Santos, M.N., Bodunov, E.N. and Valeur, B.
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| | year = 2005
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| | title = Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential)
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| | journal = [[Chemical Physics]]
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| | volume = 315
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| | pages = 171–182
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| | doi = 10.1016/j.chemphys.2005.04.006
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| |bibcode = 2005CP....315..171B }}.</ref> <math>
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| \rho (u ) = -{ 1 \over {\pi u }} \sum\limits_{k = 0}^\infty
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| {{( - 1)^k } \over {k!}}\sin (\pi \beta k)\Gamma (\beta k + 1) u^{\beta k }
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| </math>
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| Figure 2 shows the same results plotted in both a [[linear]] and a [[Logarithm|log]] representation. The curves converge to a [[Dirac delta function]] peaked at u=1 as β approaches 1, corresponding to the simple exponential function.
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| {| class="wikitable" style="margin: 1em auto 1em auto"
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| {|
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| |-
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| | [[Image:KWW dist. function linear.png|300px]] || [[Image:KWW dist. funct. log.png|300px]]
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| |}
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| |-
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| |'''Figure 2'''. Linear and log-log plots of the stretched exponential distribution function <math>G</math> vs <math>t/\tau</math>
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| for values of the stretching parameter ''β'' between 0.1 and 0.9.
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| |}
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| The moments of the original function can be expressed as
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| : <math>\langle\tau^n\rangle = \Gamma(n) \int_0^\infty d\tau\, t^{n}\, \rho(\tau)</math>.
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| The first logarithmic moment of the distribution of simple-exponential relaxation times is
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| : <math>\langle\ln\tau\rangle = \left( 1 - { 1 \over \beta } \right) {\rm Eu} + \ln \tau_K </math>
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| where Eu is the [[Euler constant]].<ref>{{cite journal | |
| | doi = 10.1063/1.1446035
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| | author = Zorn, R.
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| | year = 2002
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| | title = Logarithmic moments of relaxation time distributions
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| | journal = [[Journal of Chemical Physics]]
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| | volume = 116
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| | pages = 3204–3209
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| |bibcode = 2002JChPh.116.3204Z }}</ref>
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| == Fourier Transform ==
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| To describe results from spectroscopy or inelastic scattering, the sine or cosine Fourier transform of the stretched exponential is needed. It must be calculated either by numeric integration, or from a series expansion.<ref>Dishon et al. 1985.</ref> The series here as well as the one for the distribution function are special cases of the [[Fox-Wright function]].<ref>{{cite journal
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| | author = Hilfer, J.
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| | year = 2002
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| | title = ''H''-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems
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| | journal = [[Physical Review]] E
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| | volume = 65
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| | pages = 061510
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| }}</ref> For practical purposes, the Fourier transform may be approximated by the [[Havriliak-Negami relaxation|Havriliak-Negami function]],<ref>{{cite journal
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| | author = Alvarez, F., Alegría, A. and Colmenero, J.
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| | year = 1991
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| | title = Relationship between the time-domain Kohlrausch-Williams-Watts and frequency-domain Havriliak-Negami relaxation functions
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| | journal = [[Physical Review]] B
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| | volume = 44
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| | pages = 7306–7312
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| | doi = 10.1103/PhysRevB.44.7306
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| |bibcode = 1991PhRvB..44.7306A }}</ref>
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| though nowadays the numeric computation can be done so efficiently<ref>{{cite journal
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| | author = Wuttke, J.
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| | year = 2012
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| | title = Laplace–Fourier Transform of the Stretched Exponential Function: Analytic Error Bounds, Double Exponential Transform, and Open-Source Implementation "libkww"
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| | journal = [[Algorithms]]
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| | volume = 5
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| | pages = 604–628
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| | doi = 10.3390/a5040604}}</ref> that there is no longer any reason not to use the Kohlrausch-Williams-Watts function in the frequency domain.
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| == History and further applications ==
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| As said in the introduction, the stretched exponential was introduced by the [[Germans|German]] [[physics|physicist]] [[Rudolf Kohlrausch]] in 1854 to describe the discharge of a capacitor ([[Leyden jar]]) that used glass as dielectric medium. The next documented usage is by [[Friedrich Kohlrausch]], son of Rudolf, to describe torsional relaxation. [[A. Werner]] used it in 1907 to describe complex luminescence decays; [[Theodor Förster]] in 1949 as the fluorescence decay law of electronic energy donors.
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| Outside condensed matter physics, the stretched exponential has been used to describe the removal rates of small, stray bodies in the solar system,<ref>{{cite journal
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| | author = Dobrovolskis, A., Alvarellos, J. and Lissauer, J.
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| | year = 2007
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| | title = Lifetimes of small bodies in planetocentric (or heliocentric) orbits
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| | journal = [[Icarus (journal)|Icarus]]
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| | volume = 188
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| | pages = 481–505
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| | doi = 10.1016/j.icarus.2006.11.024|bibcode = 2007Icar..188..481D }}</ref>
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| and the diffusion-weighted MRI signal in the brain.<ref>{{cite journal
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| | author = Bennett, K. ''et al.''
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| | year = 2003
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| | title = Characterization of Continuously Distributed Water Diffusion Rates in Cerebral Cortex with a Stretched Exponential Model
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| | journal = Magn. Reson. Med.
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| | volume = 50
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| | pages = 727–734
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| | doi = 10.1002/mrm.10581}}</ref>
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| === In probability ===
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| If the integrated distribution is a stretched exponential, the normalized [[Probability distribution|probability density function]] is given by,
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| :<math>p(\tau \mid \lambda, \beta)~d\tau = \frac{\lambda}{\Gamma(1 + \beta^{-1})}~e^{-(\tau \lambda)^\beta}~d\tau</math>
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| Note that confusingly some authors<ref>{{cite book
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| | author = Sornette, D.
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| | year = 2004
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| | title = Critical Phenomena in Natural Science: Chaos, Fractals, Self-organization, and Disorder}}.</ref>
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| have been known to use the name "stretched exponential" to refer to the [[Weibull distribution]].
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| === Modified functions ===
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| A modified function
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| :<math>f_\beta (t) = e^{ -t^{\beta(t)} }</math>
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| with a slowly ''t''-dependent exponent ''β'' has been used for biological survival curves.<ref>{{cite journal
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| | author = B. M. Weon and J. H. Je
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| | year = 2009
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| | title = Theoretical estimation of maximum human lifespan
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| | journal = Biogerontology
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| | volume = 10
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| | pages = 65–71
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| | doi = 10.1007/s10522-008-9156-4}}</ref>
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| ==References==
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| <references/>
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| ==External links==
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| * J. Wuttke: [http://apps.jcns.fz-juelich.de/kww libkww] C library to compute the Fourier transform of the stretched exponential function
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| {{DEFAULTSORT:Stretched Exponential Function}}
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| [[Category:Exponentials]]
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