Specific heat for a strong-coupling superconductor with logarithmic electronic density of states

We find that a Van Hove singularity can affect the temperature (T) variation of the electronic-specific-heat difference between the superconducting and normal states \ensuremath{\Delta}C(T). It shifts the zero in this quantity away, rather than towards ${\mathit{T}}_{\mathit{c}}$ as is indicated in some experiments in the copper oxides. To carry out the calculations we have generalized the Bardeen-Stephan formula for the free-energy difference between the normal and superconducting states, and the Eliashberg equations on the imaginary-frequency axis to include explicitly a logarithmic electronic density of states (a Van Hove singularity) with normal- and paramagnetic-impurity scattering included.