= Rotated null contours for half-range Fourier inversion
{title2=$\int_{\partial E}e^{ikx}Q(\alpha^2k)\,dk=\int_{\partial D}e^{ikx}Q(\alpha k)\,dk=0$}
For $Q(k)=\int_0^\infty e^{-ikx}q(x)\,dx$, the <Half-range Fourier transform> is <holomorphic> in the lower half-plane. Let $\alpha=e^{2\pi i/3}$, $E=\{0<\arg k<\pi/3\}$ and $D=\{2\pi/3<\arg k<\pi\}$, with their boundaries oriented counterclockwise. The rotated arguments $\alpha^2k$ on $E$ and $\alpha k$ on $D$ remain in the lower half-plane. For $x>0$ and transform-admissible smooth decay, the <Cauchy integral theorem> and <Jordan lemma> give the displayed null integrals. Arbitrary multiples can therefore be added to <Fourier inversion> without changing the recovered function on the positive half-line.
// Target: integrable-systems.bigb
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