 Airy displacement integral

For $q=e^{i\pi /6}$ define $I_0(C)={\int }_0^{\infty }\mathrm{Ai}(q(s-C))ds$ and $I_1(C)={\int }_0^{\infty }s\mathrm{Ai}(q(s-C))ds$. Both converge by Airy function decay. Integrating its differential equation gives $I_1-C{I}_0=iq\mathrm{Ai}\prime (-qC)$. These moments convert a decaying vorticity profile into its velocity and displacement for three-layer long-wave shear-flow matching.