The removal of unphysical singularities in the perturbatively calculable
part of the pion form factor-a classic example of a three-point function
in QCD-is discussed. Different analytization procedures in the sense of
Shirkov and Solovtsov are examined in comparison with standard QCD
perturbation theory. We show that demanding the analyticity of the
partonic amplitude as a whole, as proposed before by Karanikas and
Stefanis, one can make infrared finite not only the strong running
coupling and its powers, but also cure potentially large logarithms
(that first appear at next-to-leading order) containing the
factorization scale and modifying the discontinuity across the cut along
the negative real axis. The scheme used here generalizes the analytic
perturbation theory of Shirkov and Solovtsov to noninteger powers of the
strong coupling and diminishes the dependence of QCD hadronic quantities
on all perturbative scheme and scale-setting parameters, including the
factorization scale
