The real and imaginary part of the vacuum polarization function
$\Pi(q^2)$ induced by a massive quark is calculated in perturbative
QCD up to order $\alpha_s^2$.
We combine the information from small and large momentum
region and from the threshold using conformal mapping
and Pad\'e approximation. This leads us to formulae for
$\Pi(q^2)$ valid for arbitrary $m^2/q^2$. Taking subsequently
the imaginary part we get the ${\cal O}(\alpha_s^2)$ to
$R \equiv \sigma(e^+ e^- \to \mbox{hadrons})/
\sigma(e^+ e^- \to \mu^+ \mu^-)$.
This extends the calculation by
K\“all\'en and Sabry from two to three loops.
