  The suitability of a double Havriliak-Negami (HN) approximant to represent
the Fourier Transform of the time derivative of Kohlrausch-Williams-Watts
function, -\psi_{\beta}, has been discussed in the first part of this work.
There, it is established the local character of the approximation and how, with
slight variation of the parameters
\{\alpha_{1,2},\gamma_{1,2},\tau_{1,2},\lambda\} with frequency, Ap_{2}HN can
describe a perfect fit with the objective function, \psi_{\beta}. Such
adiabatic behavior is commonly misunderstood as an argument against the
approximation by means of basic relaxation functions as Havriliak-Negami; this
fact it is best interpreted as the need for a wider family of relaxations with
a known local portrayal.
  Two new sets of models for describing compactly the Fourier Transform of
Kohlrausch-Williams-Watts are proposed, both based on the adiabatic variation
of parameters of a double Havriliak-Negami approximation along the whole
interval of frequencies. The first one is relying, obviously, on the use of a
well-behaved-pair of patches of the mentioned type of approximants,
\mathcal{A}p_{2}HN(\omega). The second is obtained by altering the simple
functions HN(\omega) and making dissimilar the couple. They are proposed the
guidelines of a new and systematic approach with extended Havriliak-Negami
functions which is global, (non local), and of constant parameters. The latter
at the cost of a more complicated dependency with the low frequencies than
1+(i\omega\tau_{HN})^{\alpha}.
