Pauli limiting of the upper critical magnetic field

We consider the effect of Pauli limiting on the upper critical magnetic field $H_{c2}$ of a strong-coupling superconductor with arbitrary concentration of impurities. In the extreme Pauli limit, $H_{c2P}$ is independent of the impurities but does not correspond to the highest attainable value for $H_{c2}$ except for the dirty BCS limit. In an appropriate approximation, our equations reduce correctly to the well-known Werthamer-Helfand-Hohenberg equations with an important difference that the band-splitting term is renormalized by a factor of (1+λ${)}^{\mathrm{-}1}$. This greatly reduces its effect. Strong-coupling corrections that go beyond this renormalization are also considered.