= Solution
For a fixed allowed band/subband pair, parabolic dispersions give $E_c(\mathbf k)-E_v(\mathbf k)=E_{j,n}^{\rm edge}+\hbar^2k_\parallel^2/(2\mu_j)$. If the <photon> <energy> exceeds the edge, the contour is a circle of radius $k_\parallel$ and the gradient magnitude is $\hbar^2k_\parallel/\mu_j$. Thus
$$
\int_C\frac{dl}{|\nabla_{\mathbf k}(E_c-E_v)|}=\frac{2\pi k_\parallel}{\hbar^2k_\parallel/\mu_j}=\frac{2\pi\mu_j}{\hbar^2}.
$$
Below threshold the contour is absent. With the conventional $d^2k/(2\pi)^2$ normalization, the <two-dimensional joint density of states> per area and independent channel is
$$
\boxed{J_{j,n}(\hbar\omega)=\frac{\mu_j}{2\pi\hbar^2}\Theta(\hbar\omega-E_{j,n}^{\rm edge}).}
$$
With a locally energy-independent optical <matrix element>, \b[each allowed pair therefore adds a constant step] to the continuum <dielectric function> in the approximation of the supplied equation. Different hole families need not have equal step heights: <reduced masses>, <spin> multiplicities and Bloch-state <matrix elements> differ. Frequency-dependent overall optical prefactors, nonparabolicity, broadening and excitonic Coulomb enhancement can modify the literal step shape outside this approximation.
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