Rotated null contours for half-range Fourier inversion (${\int }_{\partial E}{e}^{ikx}Q({\alpha }^{2}k)dk={\int }_{\partial D}{e}^{ikx}Q(\alpha k)dk=0$)

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 ${\alpha }^{2}k$ 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