  Recent angle-resolved photoemission (ARPES) experiments have indicated that
the electronic dispersion in some of the cuprates possesses an extended saddle
point near the Fermi level which gives rise to a density of states that
diverges like a power law instead of the weaker logarithmic divergence usually
considered. We investigate whether this strong singularity can give rise to
high transition temperatures by computing the critical temperature $T_c$ and
isotope effect coefficient $\alpha$ within a strong- coupling Eliashberg theory
which accounts for the full energy variation of the density of states. Using
band structures extracted from ARPES measurements, we demonstrate that, while
the weak-coupling solutions suggest a strong influence of the strength of the
Van Hove singularity on $T_c$ and $\alpha$, strong-coupling solutions show less
sensitivity to the singularity strength and do not support the hypothesis that
band structure effects alone can account for either the large $T_c$'s or the
different $T_c$'s within the copper oxide family. This conclusion is supported
when our results are plotted as a function of the physically relevant self-
consistent coupling constant, which show universal behavior at very strong
coupling.
