= Solution
Use a <Fourier transform> parallel to the boundary. Splitting its traces into right- and left-supported parts gives <Half-range Fourier transforms> $F_+$ and $F_-$, analytic respectively above and below a common inversion contour. A small positive imaginary part of the <wavenumber> implements the <limiting absorption principle> and separates the incident <poles> and outgoing <branch points>. Solve the transformed differential equation normal to the boundary using its outgoing or decaying solution. The boundary conditions then yield a <Wiener-Hopf equation> of the form
$$
K(\alpha)F_+(\alpha)=F_-(\alpha)+H(\alpha).
$$
Factor its <Wiener-Hopf kernel> as $K=K_+K_-$, with each factor analytic and nonzero in its designated half-plane. Divide by $K_-$ and split $H/K_-=H_++H_-$, allocating <poles> according to the incident-wave prescription. Then
$$
K_+F_+-H_+=F_-/K_-+H_-.
$$
The two sides continue to a common <entire function>. Their growth at infinity, inherited from the field's edge behavior, determines that <function>, usually a constant or a <polynomial>; in a decaying case it is zero. Solve for the unknown transforms and apply <Fourier inversion>, preserving the outgoing contour prescription.
The principal analytic results used are as follows. The <Paley–Wiener theorem> and its half-line Laplace-transform version relate support on a half-line to holomorphy in a half-plane, with growth or square-integrability bounds. The <Cauchy integral formula> and <Sokhotski–Plemelj theorem> provide additive splitting: a Cauchy <integral> of a sufficiently decaying jump has upper and lower boundary values with that prescribed difference. Applying the same construction to a continuous logarithm of a nonvanishing scalar <Wiener-Hopf kernel> gives <Wiener-Hopf factorization> when its <winding number> is zero; a nonzero index requires an explicit index factor. The <identity theorem> gives <analytic continuation> across the common strip. Finally <Liouville's theorem> makes a bounded <entire function> constant; its polynomial-growth extension bounds the possible degree. The physical edge and <radiation conditions> select the remaining constants. Zeros, <branch cuts>, nonzero index and insufficient decay must be accounted for rather than discarded during splitting.
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