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{\centering \Large \bf Reconstruction of heavy quark current correlators at ${\cal O}(\alpha_s^3)$\\}

\medskip

\noindent The files of the form ${\tt approx\_\delta\_nlX.m}$ with $\delta\; \epsilon\;
\{{\tt v,\,a,\,s,\,p\}}$ each contain one typical approximant to the vector,
axial-vector, scalar or pseudo-scalar polarisation function with
$n_l={\tt X}$ light flavours. The approximants are given in terms of
\begin{equation*}
{\tt z}=\frac{q^2}{4m^2}
\end{equation*}
and
\begin{align*} {\tt pi1[z]} =\Pi^{(1),v}(z)=&\frac{3}{16
    \pi^2}\left[\frac{5}{6}
    +\frac{13}{6z}-(1-z)\frac{3+2z}{z}G(z)+(1-z)\frac{1-16z}{6z}G(z)^2 \right.\notag\\
  &\left. -\frac{1+2z}{6z}
    \left(1+2z(1-z)\frac{d}{dz}\right)\frac{I(z)}{z}\right]\,,
\end{align*}
with
\begin{align*}
  \label{eq:14}
  I(z)=& 6\big(\zeta_3+4\Li_3(-u)+2\Li_3(u)\big)
  -8\big(2\Li_2(-u)+\Li_2(u)\big)\log\left(u\right)
  \notag\\&-2\big(2\log\left(1+u\right)+\log\left(1-u\right)\big)\log\left(u\right)^2\,,\\
  G(z) =& \frac{1}{2z}\frac{\log\left(u\right)}{\sqrt{1-\frac{1}{z}}}\,,
  \hspace{3cm} u =
  \frac{\sqrt{1-\frac{1}{z}}-1}{\sqrt{1-\frac{1}{z}}+1}\,.
\end{align*}
{\tt pi1[z]} is also given in computer readable form in the file {\tt pi1.m}
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