Cubic-dispersion elimination of one missing boundary trace ($c_1=-{\alpha }^{-2r},c_2=-{\alpha }^{-r},r=2-j$)

For the backward-sign Airy half-line global relation, suppose exactly one trace of derivative order $j\in \{0,1,2\}$ is unknown. Its spectral multiplier is a constant times $k^r$, $r=2-j$. Use the rotated null contours for half-range Fourier inversion with the displayed coefficients. The unknown contribution becomes an integral over the boundary of the middle upper sector $\pi /3<\arg k<2\pi /3$. There $\mathrm{Im}{k}^3<0$, sois bounded, while $e^{ikx}$ decays exponentially. The Cauchy integral theorem therefore removes that entire unknown contribution. The resulting solution depends on the initial transform and the two prescribed boundary transforms only.

// Target: analysis.bigb