Stretched exponential function

Extended exponential function (blue); ordinary exponential function with (black); compressed exponential function with (red). ${\displaystyle \beta ={\mbox{2/5}}=0{,}4}$ ${\displaystyle \beta =1}$ ${\displaystyle \beta ={\mbox{5/2}}=2{,}5}$

As stretched exponential called mathematical function is a generalization of the exponential function with an additional parameter in the exponent : ${\displaystyle \beta >0}$

${\displaystyle \phi (t)=e^{-\left(t/\tau \right)^{\beta }}}$

or, with : ${\displaystyle \alpha =\tau ^{-\beta }}$

${\displaystyle \phi (t)=e^{-\alpha \,t^{\beta }}}$.

In most applications , what goes hand in hand with the eponymous stretching: the function drops more slowly than the usual exponential function . For we get the compressed exponential function , for the Gaussian function . Application is, among other things, the Weibull distribution . ${\displaystyle \beta <1}$ ${\displaystyle \beta =1}$ ${\displaystyle \beta >1}$ ${\displaystyle \beta =2}$

The extended exponential function was introduced by Rudolf Kohlrausch in 1854 to describe the relaxation of the electrical polarization of a capacitor with a glass dielectric .

The stretched exponential function is also known as the Kohlrausch function or Kohlrausch-Williams-Watts function , after Graham Williams and David C. Watts , who rediscovered it in 1970.

Individual evidence

1. ↑ R. Kohlrausch: Theory of the electrical residue in the Leidner bottle. In: Annalen der Physik und Chemie Vol. 91, 1854, pp. 56–82, 179–214; online (pp. 56-82) online (pp. 179-214) .
2. ^ G. Williams, D. C. Watts: Non-Symmetrical Dielectric Relaxation Behavior Arising from a Simple Empirical Decay Function. In: Transactions of the Faraday Society Vol. 66, 1970, pp. 80-85; doi : 10.1039 / TF9706600080