Statistical Physics (phy831): Part 4: Superconductors at Finite Temperature. Introduction to Dynamics of Many Component Systems

In writing this last expression, we have used the fact that the integrand is now convergent at large x. The upper limit of the integral βch̄ωc/2 >> 1 in all known cases of superconductivity, so we take the upper limit to infinity. In evaluating the first term note that the limit x → 0 of ln(x)Tanh(x) is convergent even though ln(x) diverges as x→ 0. The integral that remains is a tabulated definite integral and its value is ln(4e/π). In the first term we also set Tanh(βh̄ωc/2)→ 1, again due to the fact that βh̄ωc/2 is large, then