  The Berger model of perturbative fragmentation of quarks to pions is improved
by providing an absolute normalization and keeping all terms in a (1-z)
expansion, which makes the calculation valid at all values of fractional pion
momentum z. We also replace the nonrelativistic wave function of a loosely
bound pion by the more realistic procedure of projecting to the light-cone pion
wave function, which in turn is taken from well known models. The full
calculation does not confirm the (1-z)^2 behavior of the fragmentation function
(FF) predicted in for $z>0.5$, and only works at very large z>0.95, where it is
in reasonable agreement with phenomenological FFs. Otherwise, we observe quite
a different z-dependence which grossly underestimates data at smaller z. The
disagreement is reduced after the addition of pions from decays of light vector
mesons, but still remains considerable. The process dependent higher twist
terms are also calculated exactly and found to be important at large z and/or
pT.
