  We investigate the correspondence between holomorphic automorphic forms on
the upper half-plane with complex weight and parabolic cocycles. For integral
weights at least 2 this correspondence is given by the Eichler integral. Knopp
generalized this to real weights. We show that for weights that are not an
integer at least 2 the generalized Eichler integral gives an injection into the
first cohomology group with values in a module of holomorphic functions, and
characterize the image. We impose no condition on the growth of the automorphic
forms at the cusps.
  For real weights that are not an integer at least 2 we similarly characterize
the space of cusp forms and the space of entire automorphic forms. We give a
relation between the cohomology classes attached to holomorphic automorphic
forms of real weight and the existence of harmonic lifts.
  A tool in establishing these results is the relation to cohomology groups
with values in modules of "analytic boundary germs", which are represented by
harmonic functions on subsets of the upper half-plane. Even for positive
integral weights cohomology with these coefficients can distinguish all
holomorphic automorphic forms, unlike the classical Eichler theory.
