  An improved approximation via Havriliak-Negami (HN) functions to the Fourier
Transform (FT) of certain Weibull distributions, -\psi_{\beta}, (the time
derivative of the Kohlrausch-Williams-Watts function), is given for a large
interval of frequencies: \omega/2\pi\in[0,10^{12}] if 0<\beta\leq1 and
\omega/2\pi\in[0,10^{7}] if 1<\beta\leq2. The model is free from the usual
numerical distortions, or restrictions associated to sampling step and finite
size, present in similar adjustments to complex relaxation functions. Further
indicates that the identification of (FT) Weibull data with a double HN
approximant, \psi_{\beta}\simeq\mathcal{A}p_{2}HN, is quite exact locally even
though the parameters involved should vary adiabatically with the frequency,
i.e. \{\alpha_{1,2},\gamma_{1,2},\tau_{1,2},\lambda\}(\omega). This fact is the
base for the high sensibility of the parameter models, as functions of \beta,
to the available data and sampling associated errors. Presumably is also the
root of the different proposals for asymptotic traits of
\alpha\cdot\gamma(\beta)_{1,2}|_{\omega\rightarrow\infty} functions found in
scientific literature.
