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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 exception of β = 2, which gives the normal ...
Depending on application, the Fourier transform of the stretched exponential function can be a viable alternative that has one parameter less. For β = 1 {\displaystyle \beta =1} the Havriliak–Negami equation reduces to the Cole–Cole equation , for α = 1 {\displaystyle \alpha =1} to the Cole–Davidson equation .
He is the co-discoverer of the Kohlrausch-Williams-Watts [KWW] stretched-exponential function for modelling molecular-relaxation processes in condensed media, [3] and has been awarded numerous awards for his research work in the field. [1]
The Cole-Cole equation's time domain current response corresponds to the Curie–von Schweidler law and the charge response corresponds to the stretched exponential function or the Kohlrausch–Williams–Watts (KWW) function, for small time arguments. [2]
Its complementary cumulative distribution function is a stretched exponential function. The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the exponential distribution (k = 1) and the Rayleigh distribution (k = 2 and = [5]).
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In 1854 Kohlrausch introduced the relaxation phenomena, and used the stretched exponential function to explain relaxation effects of a discharging Leyden jar (capacitor). [2] [3] In an 1855 experiment (published 1857) with Wilhelm Weber (1804–1891), he demonstrated that the ratio of electrostatic to electromagnetic units produced a number similar to the value of the speed of light, [4] a ...
The last image we have of Patrick Cagey is of his first moments as a free man. He has just walked out of a 30-day drug treatment center in Georgetown, Kentucky, dressed in gym clothes and carrying a Nike duffel bag.