ℓ , we need only to include the ℓ = 0 in it. Eq.(14) has the consequence that for short-range interactions, T$_{c}$/ω$_{c}$ is smaller for ℓ = 0 than for ℓ = 0, with the decrease depending on the range of interactions. Just as Eq.(14) does not include the effect of pair-breaking due to any magnetic impurities present for ℓ = 0 wave or due to any form of impurity for ℓ = 0, it does not include the effect of pair-breaking due to inelastic (real processes as opposed to virtual processes) scattering. These processes weigh the spectrum of fluctuations over the range that their frequency is smaller than O ( T ). More detailed analysis of the Eliashberg equations is required to study this. A quantity of interest to study the role of the frequency dependence of the fluctuations is the functional derivative of T$_{c}$ with respect to the change in the spectral function A ( ω ) [21] which appears in the kernel of the Eliashberg integral equations: For s-wave pairing, the analysis of Bergmann and Rainer shows that δT$_{c}$ ωδA ( ω ) > 0 for any frequency for s-wave pairing, reducing to 0 in the limit ω → 0, consistent with Anderson’s theorem about the effect of weak impurity scattering on T$_{c}$ . Increase of spectral weight in any region of the frequency spectrum increases T$_{c}$ for s-wave pairing. The maximum value of this quantity appears to occur near ω ≈ 2 πT$_{c}$ . For finite ℓ , the situation is different and depends on the ratio g = λ$_{ℓ}$/λ$_{s}$ . Then, for g < 1, one can define a frequency [23] such that for ω ≳ ω $^{∗}$, ω δT$_{c}$ δA ( ω ) ≳ 0, while for ω ≲ ω $^{∗}$, ω δT$_{c}$ δA ( ω ) ≲ 0. The physical reason for this is that thermal occupation of the fluctuations (other than those of long wavelenth which only lead to forward scattering) has similar effect as impurity scattering. This is pair-breaking for ℓ = 0 because it mixes the phase of gap at different parts of k -space in the pure limit. These conclusion are found in the solution of the Eliashberg equations for an Einstein model of spin-fluctuations, see Fig.(4). Note, also the much larger reduction in T$_{c}$ compared to the zero-temperature superconducting gap Δ(0), leading to a large value of Δ(0) /T$_{c}$ characteristic of the Cuprates. ω ∗ ≈ T$_{c}$ exp(1 /g ) (16) 21 1 ω δT$_{c}$ δA ( ω ) (15)