function $\Pi_{\mbox{\scriptsize\rm I}=1}(q^2)$ only. Power counting tells us that in this case the asymptotic behaviour of the corresponding spectral function goes as a constant i.e. the power $d$ in the r.h.s. of eq. (\[eq:asymsp\]) is $d=0$. Only one derivative is required to have a well defined correlation function. The Adler function is defined as the logarithmic derivative of $\Pi(q^2)_{\mbox{\scriptsize\rm I}=1}$ ($Q^2\equiv -q^2$, with $Q^2\ge
0$ for $q^2$–spacelike), $${{\cal A}}(Q^2)\equiv -Q^2\frac{\partial \Pi_{\mbox{\scriptsize\rm
I}=1}(q^2)}{\partial Q^2}.$$ This function, for sufficiently large values of $Q^2$ is calculable in pQCD as a power series in the running coupling constant. On the other hand the vector–isovector spectral function $\frac{1}{\pi}{\mbox{\rm Im}}\Pi_{\mbox{\scriptsize\rm I}=1}(t)$ is accessible to experimental determination from $e^{+}e^{-}$ annihilation into hadrons and from the $\tau$ hadronic decays. We then have the sum rule $${{\cal A}}(Q^2)=\int_{0}^{\infty}dt\,\frac{Q^2}{(t+Q^2)^2}\,
\frac{1}{\pi}{\mbox{\rm Im}}\Pi_{\mbox{\scriptsize\rm I}=1}(t).$$

When trying to confront this sum rule with experiment, there appears the problem that the integrand in the r.h.s. is only known experimentally from threshold up to finite values of $t$. This brings in a question of [*matching*]{} whatever is known about the low energy hadronic spectral function with its asymptotic behaviour as predicted by pQCD. We shall later on come back to this important question, after the 