= Cubic-dispersion elimination of one missing boundary trace
{title2=$c_1=-\alpha^{-2r},\quad c_2=-\alpha^{-r},\quad r=2-j$}
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 $\operatorname{Im}k^3<0$, so
$$
e^{-ik^3t}F_j(k,t)=\int_0^t e^{-ik^3(t-s)}f_j(s)\,ds
$$
is 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
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