Critical comparison between time- and frequency-domain relaxation functions

Considerable work has been performed on providing a theoretical basis for the Kohlrausch-Williams-Watts (KWW) and Havriliak-Negami (HN) relaxation functions. Because of this, several papers have examined the ``interconnection'' of these two functions. In this paper, we demonstrate that, with achievable instrumental sensitivity, these two functions are distinguishable. We further address the issue of the ``universal'' limiting power laws and the ability to obtain the exponents associated with them. Finally, the stability and accuracy of our numerical Laplace transform is demonstrated by comparison between functions with known analytical time and frequency solutions. The stability of our algorithm indicates that the method of Alvarez and co-workers [F. Alvarez, A. Alegr\'{}a, and J. Colmenero, Phys. Rev. B 44, 7306 (1991)] is an unnecessary approximation for converting between the time and frequency domain.