  Analyzing the single inclusive annihilation spectrum of charged hadrons in
$e^+e^-$ collisions, I confront the hadronization hypothesis of local
parton-hadron duality with a systematic resummation of the dependence on the
small energy fraction. This resummation is based on the reciprocity between
time-like and space-like splitting processes in $4-2\epsilon$-dimensions, which
I extend to resum all the soft terms of the cross-section for inclusive jet
production. Under the local-parton-hadron duality hypothesis, the resulting
distribution of jets essentially determines the spectrum of hadrons as the jet
radius goes to zero. Thus I take the resummed perturbative jet function as the
non-perturbative fragmentation function with an effective infra-red coupling. I
find excellent agreement with data, and comment on the mixed leading log
approximation previously used to justify local parton-hadron duality.
