Stable law densities and linear relaxation phenomena

Abstract

Stable law distributions occur in the description of the linear dielectric behavior of polymers, the motion of carriers in semi-conductors, the statistical behavior of neurons, and many other phenomena. No accurate tables of these distributions or algorithms for estimating the parameters in these relaxation models exist. In this paper we present tables of the functions $$\begin{gathered} Q_{\alpha }(\mathrm{z})=\frac{1}{\pi }{\int }_0^{\infty }{e}^{-u^{\alpha }}\cos (zu)du \\ {V}_{\alpha }(z)=\frac{1}{\pi }{\int }_0^{\infty }{e}^{-u^{\alpha }}\sin (zu)du \end{gathered}$$ together with related functional properties of zQ_α(z). These are useful in the estimation of the parameters in relaxation models for polymers and related materials. Values of the integral Q_α(z) are given for α = 0.01,0.02(0.02)0.1(0.1)1.0(0.2)2.0 and those of V_α(z) are given for α = 0.0(0.01)0.1(0.1)2.0. A variety of methods was used to obtain six place accuracy. The tables can be used to sequentially estimate the three parameters appearing in the Williams-Watts model of relaxation. An illustration of this method applied to data in the literature is given.