Gravitational instability of an incompressible slab ($kH<q_c$)

The even surface branch is unstable for $0<kH<q_c$, where $q_c=(1+e^{-2q_c})/2\simeq 0.6392322714$, and stable above this threshold. All positive-wavenumber odd modes are stable because $1-e^{-2q}<2q$. The equilibrium nevertheless has a long-wavelength column-density instability.