= Cubic Stokes dispersion symmetry
{title2=$\nu_{1,2}=(-k\pm\sqrt{4-3k^2})/2$}
For the <linear dispersive Stokes equation>, $\omega(\nu)=\omega(k)$ factors as $(\nu-k)(\nu^2+k\nu+k^2-1)=0$. In $D_+=\{\operatorname{Im}k>0,\operatorname{Re}\omega(k)<0\}$ the two displayed other roots lie in the lower half-plane, so both can be used in the <global relation>. To check the root locations, at $k=ib$ they have imaginary part $-b/2$. A root could reach the real axis only where $\operatorname{Re}\omega(k)=0$, so continuity preserves their signs throughout this connected domain. The branch points $\pm2/\sqrt3$ lie outside its closure. The weights $A_1=(k-\nu_2)/(\nu_1-\nu_2)$ and $A_2=(\nu_1-k)/(\nu_1-\nu_2)$ obey $A_1+A_2=1$ and $\nu_j'=-A_j$.
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