= Half-line drift null contour identity
{title2=$\int_L e^{ikx}\widehat q(i\alpha-k)\,dk=0$}
For $\alpha>0$, let $L$ be the upper branch $\operatorname{Im}k=(\alpha+\sqrt{\alpha^2+4(\operatorname{Re}k)^2})/2$, oriented left to right. Above $L$, $i\alpha-k$ lies in the lower half-plane, where a decaying half-line <Fourier transform> is analytic. For $x>0$, closing upward proves that the displayed integral vanishes. An additional multiplier analytic above $L$ with at most polynomial growth is allowed, using exponential decay and, if needed, regularized contours. This supplies a null term in a <Fokas method> inversion formula without changing the function in the interior.
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