  The basic theoretical foundation for the modelling of phonon assisted
absorption spectra in direct bandgap semiconductors, introduced by Elliott 60
years ago using second order perturbation theory, results in a square root
shaped dependency close to the absorption edge. A careful analysis of the
experiments reveals that for the yellow S excitons in Cu$_2$O the lineshape
does not follow that square root dependence. The reexamination of the theory
shows that the basic assumptions of constant matrix elements and constant
energy denominators is invalid for semiconductors with dominant exciton effects
like Cu$_2$O, where the phonon assisted absorption proceeds via intermediate
exciton states. The overlap between these and the final exciton states strongly
determines the dependence of the absorption on the photon energy. To describe
the experimental observed line shape of the indirect absorption of the yellow S
exciton states we find it necessary to assume a momentum dependent deformation
potential for the optical phonons.
