Half-range Fourier transform ($F^{\pm }(k)={\int }_{{\mathbb{R}}_{\pm }}{e}^{ikx}f(x)dx$)

With the $e^{ikx}$ convention, a right-supported decaying function has a transform analytic in the upper half-plane and a left-supported one in the lower half-plane. Incident modes continued with small imaginary frequency can have poles outside the initial convergence strip; these poles must be tracked separately from outgoing scattered waves.

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

For $\alpha >0$, let $L$ be the upper branch $\mathrm{Im}k=(\alpha +\sqrt{{\alpha }^2+4(\mathrm{Re}k{)}^2})/2$, 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.