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Stretched Exponential Function

The stretched exponential function, also called the Kohlrausch or Kohlrausch-Williams-Watts function, is the function

f_(alpha,tau_alpha)(t)=exp[-(t/(tau_alpha))^alpha],
(1)

where t>=0, tau_alpha>0, and conventionally 0<alpha<=1 (Williams and Watts 1970). The case alpha=1 is the ordinary exponential function.

For 0<alpha<=1, the function is completely monotonic and hence is a Laplace transform. More explicitly,

e^(-x^alpha)=int_0^inftye^(-rx)k_alpha(r)dr,
(2)

where k_alpha is the one-sided stable distribution with stability index alpha (Pollard 1946). Consequently, its exact probability density over exponential relaxation times is

f_(alpha,tau_alpha)(t)=int_0^inftye^(-t/tau)G_alpha(tau)dtau
(3)
G_alpha(tau)=(tau_alpha)/(tau^2)k_alpha((tau_alpha)/tau).
(4)

For example, when alpha=1/2,

G_(1/2)(tau)=(exp[-tau/(4tau_alpha)])/(2sqrt(pitau_alphatau)).
(5)

The improper integral of the stretched exponential, also interpreted as its mean relaxation time, is

int_0^inftyf_(alpha,tau_alpha)(t)dt=tau_alphaGamma(1+1/alpha).
(6)

This should not be confused with using a normalized stretched exponential itself as a phenomenological probability density,

g_alpha(tau)=(exp[-(tau/tau_alpha)^alpha])/(tau_alphaGamma(1+1/alpha)),
(7)

which is generally different from the exact relaxation-time probability density G_alpha (Ribeiro de Almeida et al. 2026).

See also

Completely Monotonic Function, Exponential Function, Laplace Transform, Stable Distribution

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References

Pollard, H. "The Representation of e^(-x^lambda) as a Laplace Integral." Bull. Amer. Math. Soc. 52, 908-910, 1946. https://doi.org/10.1090/S0002-9904-1946-08672-3.Ribeiro de Almeida, R. R.; Lenzi, E. K.; and Evangelista, L. R. "Impedance Response of Electrolytic Cells with Non-Debye Relaxation Time Distributions at Adsorbing Electrodes." Electrochim. Acta 577, 149724, 2026. https://doi.org/10.1016/j.electacta.2026.149724.Williams, G. and Watts, D. C. "Non-Symmetrical Dielectric Relaxation Behaviour Arising from a Simple Empirical Decay Function." Trans. Faraday Soc. 66, 80-85, 1970. https://doi.org/10.1039/TF9706600080.

Cite this as:

Weisstein, Eric W. "Stretched Exponential Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StretchedExponentialFunction.html

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