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.