SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Stretched exponential function</title>

</head>

Stretched exponential function

</h1>

</header>

The stretched exponential function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\beta }(t)=e^{-t^{\beta }}}">

<msub>

</mrow>

</msub>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{\beta }(t)=e^{-t^{\beta }}}</annotation>

</semantics>

</math> is obtained by inserting a fractional <a href="Power_law" title="Power law">power law</a> into the <a href="Exponential_function" title="Exponential function">exponential function</a>. 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, hence the name of the function. The compressed exponential function (with β > 1) has less practical importance, with the notable exceptions of β = 2, which gives the <a href="Normal_distribution" title="Normal distribution">normal distribution</a>, and of compressed exponential relaxation in the dynamics of <a href="Amorphous_solids" class="mw-redirect" title="Amorphous solids">amorphous solids</a>.<a href="#cite_note-1">[1]</a>

In mathematics, the stretched exponential is also known as the <a href="Cumulative_distribution_function#Complementary_cumulative_distribution_function_(tail_distribution)" title="Cumulative distribution function">complementary cumulative</a> <a href="Weibull_distribution" title="Weibull distribution">Weibull distribution</a>. The stretched exponential is also the <a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a>, basically the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a>, of the <a href="Stable_distribution" title="Stable distribution">Lévy symmetric alpha-stable distribution</a>.

In physics, the stretched exponential function is often used as a phenomenological description of <a href="Relaxation_(physics)" title="Relaxation (physics)">relaxation</a> in disordered systems. It was first introduced by <a href="Rudolf_Kohlrausch" title="Rudolf Kohlrausch">Rudolf Kohlrausch</a> in 1854 to describe the discharge of a capacitor;<a href="#cite_note-2">[2]</a> thus it is also known as the Kohlrausch function. In 1970, G. Williams and D.C. Watts used the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of the stretched exponential to describe <a href="Dielectric_spectroscopy" title="Dielectric spectroscopy">dielectric spectra</a> of polymers;<a href="#cite_note-3">[3]</a> in this context, the stretched exponential or its Fourier transform are also called the Kohlrausch–Williams–Watts (KWW) function. The Kohlrausch–Williams–Watts (KWW) function corresponds to the time domain charge response of the main dielectric models, such as the <a href="Cole%E2%80%93Cole_equation" title="Cole–Cole equation">Cole–Cole equation</a>, the <a href="Cole%E2%80%93Davidson_equation" title="Cole–Davidson equation">Cole–Davidson equation</a>, and the <a href="Havriliak%E2%80%93Negami_relaxation" title="Havriliak–Negami relaxation">Havriliak–Negami relaxation</a>, for small time arguments.<a href="#cite_note-4">[4]</a>

In phenomenological applications, it is often not clear whether the stretched exponential function should be used to describe the differential or the integral distribution function—or neither. 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,<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a> it can be shown that the asymptotic decay is a stretched exponential, but the prefactor is usually an unrelated power.

<div class="mw-heading mw-heading2"><h2 id="Mathematical_properties">Mathematical properties</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Moments">Moments</h3></div>

Following the usual physical interpretation, we interpret the function argument t as time, and fβ(t) is the differential distribution. The area under the curve can thus be interpreted as a mean relaxation time. One finds

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \tau \rangle \equiv \int _{0}^{\infty }dt\,e^{-(t/\tau _{K})^{\beta }}={\tau _{K} \over \beta }\Gamma {\left({\frac {1}{\beta }}\right)}}</annotation>

</semantics>

</math>

where Γ is the <a href="Gamma_function" title="Gamma function">gamma function</a>. For <a href="Exponential_decay" title="Exponential decay">exponential decay</a>, ⟨τ⟩ = τK is recovered.

The higher <a href="Moment_(mathematics)" title="Moment (mathematics)">moments</a> of the stretched exponential function are<a href="#cite_note-Zwillinger_2014-9">[9]</a>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<mfrac>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \tau ^{n}\rangle \equiv \int _{0}^{\infty }dt\,t^{n-1}\,e^{-(t/\tau _{K})^{\beta }}={{\tau _{K}}^{n} \over \beta }\Gamma {\left({\frac {n}{\beta }}\right)}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Distribution_function">Distribution function</h3></div>

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

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{-t^{\beta }}=\int _{0}^{\infty }du\,\rho (u)\,e^{-t/u}.}</annotation>

</semantics>

</math>

Alternatively, a distribution <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=u\rho (u)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=u\rho (u)}</annotation>

</semantics>

</math> is used.

ρ can be computed from the series expansion:<a href="#cite_note-10">[10]</a>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho (u)=-{1 \over \pi u}\sum _{k=0}^{\infty }{(-1)^{k} \over k!}\sin(\pi \beta k)\Gamma (\beta k+1)u^{\beta k}}</annotation>

</semantics>

</math>

For rational values of β, ρ(u) can be calculated in terms of elementary functions. But the expression is in general too complex to be useful except for the case β = 1/2 where

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(u)=u\rho (u)={1 \over 2{\sqrt {\pi }}}{\sqrt {u}}e^{-u/4}}</annotation>

</semantics>

</math>

Figure 2 shows the same results plotted in both a <a href="Linear" class="mw-redirect" title="Linear">linear</a> and a <a href="Logarithm" title="Logarithm">log</a> representation. The curves converge to a <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a> peaked at u = 1 as β approaches 1, corresponding to the simple exponential function.

<td>

</td></tr>

<tr>

<td colspan="2">Figure 2. Linear and log-log plots of the stretched exponential distribution function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> vs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t/\tau }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t/\tau }</annotation>

</semantics>

</math><img src="./db5b1489e333735f1976883766ef82d909610b88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.204ex; height:2.843ex;" alt="{\displaystyle t/\tau }" loading="lazy">

for values of the stretching parameter β between 0.1 and 0.9.

</td></tr></tbody></table>

The moments of the original function can be expressed as

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \tau ^{n}\rangle =\Gamma (n)\int _{0}^{\infty }d\tau \,t^{n}\,\rho (\tau ).}</annotation>

</semantics>

</math>

The first logarithmic moment of the distribution of simple-exponential relaxation times is

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \ln \tau \rangle =\left(1-{1 \over \beta }\right){\rm {Eu}}+\ln \tau _{K}}</annotation>

</semantics>

</math>

where Eu is the <a href="Euler_constant" class="mw-redirect" title="Euler constant">Euler constant</a>.<a href="#cite_note-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Fourier_transform">Fourier transform</h2></div>

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.<a href="#cite_note-12">[12]</a> The series here as well as the one for the distribution function are special cases of the <a href="Fox%E2%80%93Wright_function" title="Fox–Wright function">Fox–Wright function</a>.<a href="#cite_note-13">[13]</a> For practical purposes, the Fourier transform may be approximated by the <a href="Havriliak%E2%80%93Negami_relaxation" title="Havriliak–Negami relaxation">Havriliak–Negami function</a>,<a href="#cite_note-14">[14]</a> though nowadays the numeric computation can be done so efficiently<a href="#cite_note-15">[15]</a> that there is no longer any reason not to use the Kohlrausch–Williams–Watts function in the frequency domain.

<div class="mw-heading mw-heading2"><h2 id="History_and_further_applications">History and further applications</h2></div>

As said in the introduction, the stretched exponential was introduced by the <a href="Germans" title="Germans">German</a> <a href="Physicist" title="Physicist">physicist</a> <a href="Rudolf_Kohlrausch" title="Rudolf Kohlrausch">Rudolf Kohlrausch</a> in 1854 to describe the discharge of a capacitor (<a href="Leyden_jar" title="Leyden jar">Leyden jar</a>) that used glass as dielectric medium. The next documented usage is by <a href="Friedrich_Kohlrausch_(physicist)" title="Friedrich Kohlrausch (physicist)">Friedrich Kohlrausch</a>, son of Rudolf, to describe torsional relaxation. A. Werner used it in 1907 to describe complex luminescence decays; <a href="Theodor_F%C3%B6rster" title="Theodor Förster">Theodor Förster</a> in 1949 as the fluorescence decay law of electronic energy donors.

Outside <a href="Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>, the stretched exponential has been used to describe the removal rates of small, stray bodies in the solar system,<a href="#cite_note-17">[17]</a> the diffusion-weighted MRI signal in the brain,<a href="#cite_note-18">[18]</a> and the production from unconventional gas wells.<a href="#cite_note-19">[19]</a>

<div class="mw-heading mw-heading3"><h3 id="In_probability">In probability</h3></div>

If the integrated distribution is a stretched exponential, the normalized <a href="Probability_distribution" title="Probability distribution">probability density function</a> is given by

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(\tau \mid \lambda ,\beta )~d\tau ={\frac {\lambda }{\Gamma (1+\beta ^{-1})}}~e^{-(\tau \lambda )^{\beta }}~d\tau }</annotation>

</semantics>

</math>

Note that confusingly some authors have been known to use the name "stretched exponential" to refer to the <a href="Weibull_distribution" title="Weibull distribution">Weibull distribution</a>.<a href="#cite_note-20">[20]</a>

<div class="mw-heading mw-heading3"><h3 id="Modified_functions">Modified functions</h3></div>

A modified stretched exponential function

<msub>

</mrow>

</msub>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{\beta }(t)=e^{-t^{\beta (t)}}}</annotation>

</semantics>

</math>

with a slowly t-dependent exponent β has been used for biological survival curves.<a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a>

<div class="mw-heading mw-heading3"><h3 id="Wireless_communications">Wireless communications</h3></div>

In wireless communications, a scaled version of the stretched exponential function has been shown to appear in the Laplace Transform for the interference power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> when the transmitters' locations are modeled as a 2D <a href="Poisson_point_process" title="Poisson point process">Poisson Point Process</a> with no exclusion region around the receiver.<a href="#cite_note-23">[23]</a>

The <a href="Laplace_transform" title="Laplace transform">Laplace transform</a> can be written for arbitrary <a href="Fading" title="Fading">fading</a> distribution as follows:

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

<mrow>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{I}(s)=\exp \left(-\pi \lambda \mathbb {E} {\left[g^{\frac {2}{\eta }}\right]}\Gamma {\left(1-{\frac {2}{\eta }}\right)}s^{\frac {2}{\eta }}\right)=\exp \left(-ts^{\beta }\right)}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> is the power of the fading, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

</semantics>

</math><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> is the <a href="Path_loss#Loss_exponent" title="Path loss">path loss exponent</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> is the density of the 2D Poisson Point Process, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (\cdot )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (\cdot )}</annotation>

</semantics>

</math><img src="./fb56054f7834e05862ff50930fc46b4f8e3f8280.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.909ex; height:2.843ex;" alt="{\displaystyle \Gamma (\cdot )}" loading="lazy"> is the Gamma function, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [x]}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [x]}</annotation>

</semantics>

</math><img src="./c8db5ff6ad5108d60a241854a6789d2b069c8662.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.174ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [x]}" loading="lazy"> is the expectation of the variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">.

The same reference also shows how to obtain the inverse Laplace Transform for the stretched exponential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp \left(-s^{\beta }\right)}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp \left(-s^{\beta }\right)}</annotation>

</semantics>

</math><img src="./555277304d48e6522a4858fb8a0f0fad410e5065.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.755ex; height:3.343ex;" alt="{\displaystyle \exp \left(-s^{\beta }\right)}" loading="lazy"> for higher order integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =\beta _{q}\beta _{b}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta =\beta _{q}\beta _{b}}</annotation>

</semantics>

</math><img src="./1abbc912cb41d17377898e3cc67e04143e4dd2f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.988ex; height:2.843ex;" alt="{\displaystyle \beta =\beta _{q}\beta _{b}}" loading="lazy"> from lower order integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{a}}</annotation>

</semantics>

</math><img src="./7d48576ab8a7b331bf471250cb91846ec78a7243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.509ex;" alt="{\displaystyle \beta _{a}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{b}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta _{b}}</annotation>

</semantics>

</math><img src="./9ebb7b4cea705dfdfe4b4db9e02303189bae9d9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.253ex; height:2.509ex;" alt="{\displaystyle \beta _{b}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Internet_streaming">Internet streaming</h3></div>

The stretched exponential has been used to characterize Internet media accessing patterns, such as YouTube and other stable <a href="Streaming_media" title="Streaming media">streaming media</a> sites.<a href="#cite_note-24">[24]</a> The commonly agreed power-law accessing patterns of Web workloads mainly reflect text-based content Web workloads, such as daily updated news sites.<a href="#cite_note-25">[25]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFTrachenkoZaccone2021" class="citation journal cs1">Trachenko, K.; Zaccone, A. (2021-06-14). <a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1088/1361-648X/ac04cd">"Slow stretched-exponential and fast compressed-exponential relaxation from local event dynamics"</a>. Journal of Physics: Condensed Matter. 33: 315101. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2010.10440">2010.10440</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1361-648X%2Fac04cd">10.1088/1361-648X/ac04cd</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0953-8984">0953-8984</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFKohlrausch,_R.1854" class="citation journal cs1">Kohlrausch, R. (1854). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k15176w.pagination">"Theorie des elektrischen Rückstandes in der Leidner Flasche"</a>. <a href="Annalen_der_Physik_und_Chemie" class="mw-redirect" title="Annalen der Physik und Chemie">Annalen der Physik und Chemie</a>. 91 (1): 56–82, 179–213. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1854AnP...167...56K">1854AnP...167...56K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.18541670103">10.1002/andp.18541670103</a>.</cite>.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFWilliams,_G.Watts,_D._C.1970" class="citation journal cs1">Williams, G. & Watts, D. C. (1970). "Non-Symmetrical Dielectric Relaxation Behavior Arising from a Simple Empirical Decay Function". Transactions of the Faraday Society. 66: 80–85. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1039%2Ftf9706600080">10.1039/tf9706600080</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:95007734">95007734</a>.</cite>.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFHolm2020" class="citation journal cs1">Holm, Sverre (2020). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7851980">"Time domain characterization of the Cole-Cole dielectric model"</a>. Journal of Electrical Bioimpedance. 11 (1): 101–105. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2478%2Fjoeb-2020-0015">10.2478/joeb-2020-0015</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7851980">7851980</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/33584910">33584910</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFDonsker,_M._D.Varadhan,_S._R._S.1975" class="citation journal cs1">Donsker, M. D. & Varadhan, S. R. S. (1975). "Asymptotic evaluation of certain Markov process expectations for large time". Comm. Pure Appl. Math. 28: 1–47. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fcpa.3160280102">10.1002/cpa.3160280102</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFTakano,_H._and_Nakanishi,_H._and_Miyashita,_S.1988" class="citation journal cs1">Takano, H. and Nakanishi, H. and Miyashita, S. (1988). "Stretched exponential decay of the spin-correlation function in the kinetic Ising model below the critical temperature". Phys. Rev. B. 37 (7): 3716–3719. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988PhRvB..37.3716T">1988PhRvB..37.3716T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.37.3716">10.1103/PhysRevB.37.3716</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9944981">9944981</a>.</cite><code class="cs1-code">{{cite journal}}</code>: CS1 maint: multiple names: authors list (link)

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFShore,_John_E._and_Zwanzig,_Robert1975" class="citation journal cs1">Shore, John E. and Zwanzig, Robert (1975). "Dielectric relaxation and dynamic susceptibility of a one-dimensional model for perpendicular-dipole polymers". The Journal of Chemical Physics. 63 (12): 5445–5458. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1975JChPh..63.5445S">1975JChPh..63.5445S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.431279">10.1063/1.431279</a>.</cite><code class="cs1-code">{{cite journal}}</code>: CS1 maint: multiple names: authors list (link)

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFBrey,_J._J._and_Prados,_A.1993" class="citation journal cs1">Brey, J. J. and Prados, A. (1993). "Stretched exponential decay at intermediate times in the one-dimensional Ising model at low temperatures". Physica A. 197 (4): 569–582. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993PhyA..197..569B">1993PhyA..197..569B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0378-4371%2893%2990015-V">10.1016/0378-4371(93)90015-V</a>.</cite><code class="cs1-code">{{cite journal}}</code>: CS1 maint: multiple names: authors list (link)

</li>

<li id="cite_note-Zwillinger_2014-9"><a href="#cite_ref-Zwillinger_2014_9-0">^</a> <cite id="CITEREFGradshteynRyzhikGeronimusTseytlin2015" class="citation book cs1"><a href="Izrail_Solomonovich_Gradshteyn" class="mw-redirect" title="Izrail Solomonovich Gradshteyn">Gradshteyn, Izrail Solomonovich</a>; <a href="Iosif_Moiseevich_Ryzhik" class="mw-redirect" title="Iosif Moiseevich Ryzhik">Ryzhik, Iosif Moiseevich</a>; <a href="Yuri_Veniaminovich_Geronimus" class="mw-redirect" title="Yuri Veniaminovich Geronimus">Geronimus, Yuri Veniaminovich</a>; <a href="Michail_Yulyevich_Tseytlin" class="mw-redirect" title="Michail Yulyevich Tseytlin">Tseytlin, Michail Yulyevich</a>; Jeffrey, Alan (2015) [October 2014]. "3.478.". In Zwillinger, Daniel; <a href="Victor_Hugo_Moll" class="mw-redirect" title="Victor Hugo Moll">Moll, Victor Hugo</a> (eds.). <a href="Gradshteyn_and_Ryzhik" title="Gradshteyn and Ryzhik">Table of Integrals, Series, and Products</a>. Translated by Scripta Technica, Inc. (8 ed.). <a href="Academic_Press%2C_Inc." class="mw-redirect" title="Academic Press, Inc.">Academic Press, Inc.</a> p. 372. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-384933-5</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2014010276">2014010276</a>.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFLindsey,_C._P.Patterson,_G._D.1980" class="citation journal cs1">Lindsey, C. P. & Patterson, G. D. (1980). "Detailed comparison of the Williams-Watts and Cole-Davidson functions". <a href="Journal_of_Chemical_Physics" class="mw-redirect" title="Journal of Chemical Physics">Journal of Chemical Physics</a>. 73 (7): 3348–3357. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1980JChPh..73.3348L">1980JChPh..73.3348L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.440530">10.1063/1.440530</a>.</cite>.

For a more recent and general discussion, see <cite id="CITEREFBerberan-Santos,_M.N.,_Bodunov,_E.N._and_Valeur,_B.2005" class="citation journal cs1">Berberan-Santos, M.N., Bodunov, E.N. and Valeur, B. (2005). "Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential)". <a href="Chemical_Physics" class="mw-redirect" title="Chemical Physics">Chemical Physics</a>. 315 (1–2): 171–182. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005CP....315..171B">2005CP....315..171B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.chemphys.2005.04.006">10.1016/j.chemphys.2005.04.006</a>.</cite><code class="cs1-code">{{cite journal}}</code>: CS1 maint: multiple names: authors list (link).

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFZorn,_R.2002" class="citation journal cs1">Zorn, R. (2002). <a rel="nofollow" class="external text" href="http://juser.fz-juelich.de/record/1954/files/10418.pdf">"Logarithmic moments of relaxation time distributions"</a> (PDF). <a href="Journal_of_Chemical_Physics" class="mw-redirect" title="Journal of Chemical Physics">Journal of Chemical Physics</a>. 116 (8): 3204–3209. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002JChPh.116.3204Z">2002JChPh.116.3204Z</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1446035">10.1063/1.1446035</a>.</cite>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> Dishon et al. 1985.

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFHilfer,_J.2002" class="citation journal cs1">Hilfer, J. (2002). "H-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems". Physical Review E. 65 (6): 061510. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002PhRvE..65f1510H">2002PhRvE..65f1510H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreve.65.061510">10.1103/physreve.65.061510</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/12188735">12188735</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16276298">16276298</a>.</cite>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFAlvarez,_F.,_Alegría,_A._and_Colmenero,_J.1991" class="citation journal cs1">Alvarez, F., Alegría, A. and Colmenero, J. (1991). "Relationship between the time-domain Kohlrausch-Williams-Watts and frequency-domain Havriliak-Negami relaxation functions". Physical Review B. 44 (14): 7306–7312. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991PhRvB..44.7306A">1991PhRvB..44.7306A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.44.7306">10.1103/PhysRevB.44.7306</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9998642">9998642</a>.</cite><code class="cs1-code">{{cite journal}}</code>: CS1 maint: multiple names: authors list (link)

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFWuttke,_J.2012" class="citation journal cs1">Wuttke, J. (2012). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fa5040604">"Laplace–Fourier Transform of the Stretched Exponential Function: Analytic Error Bounds, Double Exponential Transform, and Open-Source Implementation "libkww""</a>. <a href="Algorithms_(journal)" title="Algorithms (journal)">Algorithms</a>. 5 (4): 604–628. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0911.4796">0911.4796</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fa5040604">10.3390/a5040604</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15030084">15030084</a>.</cite>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFSluchSomozaBerg2002" class="citation journal cs1">Sluch, Mikhail I.; Somoza, Mark M.; Berg, Mark A. (1 July 2002). "Friction on Small Objects and the Breakdown of Hydrodynamics in Solution: Rotation of Anthracene in Poly(isobutylene) from the Small-Molecule to Polymer Limits". The Journal of Physical Chemistry B. 106 (29): 7385–7397. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fjp025549u">10.1021/jp025549u</a>.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFDobrovolskis,_A.,_Alvarellos,_J._and_Lissauer,_J.2007" class="citation journal cs1">Dobrovolskis, A., Alvarellos, J. and Lissauer, J. (2007). "Lifetimes of small bodies in planetocentric (or heliocentric) orbits". <a href="Icarus_(journal)" title="Icarus (journal)">Icarus</a>. 188 (2): 481–505. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007Icar..188..481D">2007Icar..188..481D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.icarus.2006.11.024">10.1016/j.icarus.2006.11.024</a>.</cite><code class="cs1-code">{{cite journal}}</code>: CS1 maint: multiple names: authors list (link)

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFBennett,_K.2003" class="citation journal cs1">Bennett, K.; et al. (2003). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fmrm.10581">"Characterization of Continuously Distributed Water Diffusion Rates in Cerebral Cortex with a Stretched Exponential Model"</a>. Magn. Reson. Med. 50 (4): 727–734. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fmrm.10581">10.1002/mrm.10581</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/14523958">14523958</a>.</cite>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite id="CITEREFValkoLee2010" class="citation journal cs1">Valko, Peter P.; Lee, W. John (2010-01-01). "A Better Way To Forecast Production From Unconventional Gas Wells". SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2118%2F134231-ms">10.2118/134231-ms</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781555633004</bdi>.</cite>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite id="CITEREFSornette,_D.2004" class="citation book cs1">Sornette, D. (2004). Critical Phenomena in Natural Science: Chaos, Fractals, Self-organization, and Disorder.</cite>.

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFB._M._WeonJ._H._Je2009" class="citation journal cs1">B. M. Weon & J. H. Je (2009). "Theoretical estimation of maximum human lifespan". Biogerontology. 10 (1): 65–71. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10522-008-9156-4">10.1007/s10522-008-9156-4</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18560989">18560989</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8554128">8554128</a>.</cite>

</li>

<cite id="CITEREFB._M._Weon2016" class="citation journal cs1">B. M. Weon (2016). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4726238">"Tyrannosaurs as long-lived species"</a>. Scientific Reports. 6: 19554. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016NatSR...619554W">2016NatSR...619554W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fsrep19554">10.1038/srep19554</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4726238">4726238</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26790747">26790747</a>.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFAmmar,_H._A.,_Nasser,_Y._and_Artail,_H.2018" class="citation book cs1">Ammar, H. A., Nasser, Y. and Artail, H. (2018). "Closed Form Expressions for the Probability Density Function of the Interference Power in PPP Networks". 2018 IEEE International Conference on Communications (ICC). pp. 1–6. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1803.10440">1803.10440</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FICC.2018.8422214">10.1109/ICC.2018.8422214</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-5386-3180-5</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4374550">4374550</a>.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFLei_Guo,_Enhua_Tan,_Songqing_Chen,_Zhen_Xiao,_and_Xiaodong_Zhang2008" class="citation conference cs1">Lei Guo, Enhua Tan, Songqing Chen, Zhen Xiao, and Xiaodong Zhang (2008). "The Stretched Exponential Distribution of Internet Media Access Patterns". PODC' 08. pp. 283–294. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1400751.1400789">10.1145/1400751.1400789</a>.</cite><code class="cs1-code">{{cite conference}}</code>: CS1 maint: multiple names: authors list (link)

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite id="CITEREFAdamicBernardo_A.2000" class="citation journal cs1">Adamic, Lada A.; Bernardo A., Huberman (2000). "Power-Law Distribution of the World Wide Web". Science. 287 (5461): 2115–2115. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.287.5461.2115a">10.1126/science.287.5461.2115a</a>.</cite>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li>J. Wuttke: <a rel="nofollow" class="external text" href="http://apps.jcns.fz-juelich.de/kww">libkww</a> C library to compute the Fourier transform of the stretched exponential function</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-24" href="https://en.wikipedia.org/wiki/?title=Stretched_exponential_function&oldid=1302289634">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>