= Logarithmic two-dimensional saddle joint density of states
{title2=$J(E)\sim g_s m^*/(2\pi^2\hbar^2)\ln(E_\Lambda/|E-E_s|)$}
A single two-dimensional quadratic saddle with $A=\hbar^2/(2m^*)$ contributes this leading <joint density of states> when both contour branches are included and $g_s$ counts independent spin channels. The contour integral is proportional to $\int_{-\Lambda}^{\Lambda}dy/\sqrt{y^2+\delta/A}=2\operatorname{arsinh}(\Lambda\sqrt{A/\delta})$ above the saddle. Its asymptote gives the logarithm with $E_\Lambda$ of order $A\Lambda^2$. The cutoff-dependent constant is not universal. This is a density per area per energy; changing to a density per frequency requires a Jacobian. <Optical Properties of Solids by Mildred S. Dresselhaus>[Chapter4, especiallyTable4.1: two-dimensional saddle critical point]{ref}
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