  Using a full resummation of the Adler function in the large-$\beta_0$
approximation of QCD and a mathematical framework of resurgence suitable for
the specific properties of the Borel transform in this particular case, we
derive a compact resurgent representation of the QCD Adler function, valid in
the whole complex momentum plane. The representation is expressed in terms of
the inverse Mellin transform of the Borel function and is analytic in the
complex momentum plane, except for cuts along the timelike axis and the Landau
region of the spacelike axis. It contains a purely nonperturbative term
singular at the origin of the coupling plane, depending on a single real
arbitrary constant. We compare the resurgent Adler function derived in this
work with a previous determination in a similar framework and use its values in
the complex plane for a calculation of the hadronic width of the $\tau$ lepton
in the Standard Model.
