Half-range Fourier transform
= Half-range Fourier transform
{c}
{title2=$F^\pm(k)=\int_{\mathbb R_\pm}e^{ikx}f(x)dx$}
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.