  A generalization of the Abrikosov-Gorkov equations for non-magnetic
impurities in unconventional superconductors is proposed, including higher
harmonics in the expansion of the momentum dependent gap function and a
momentum dependent impurity scattering potential. This model is treated within
a self-consistent calculation to obtain the electronic density of states, the
optical conductivity, and the gap function in a two-dimensional
d_{x^2-y^2}-wave superconductor. It is argued that momentum dependent
scattering from the impurities may lead to extended gapless regions in the gap
function centered around the nodes of the pure d_{x^2-y^2}-wave superconductor.
The associated enhancement of the residual density of states may be responsible
for the rapid decrease of T_c and the increase of the London penetration depth
with hole doping observed in overdoped cuprate superconductors.
