= Mixed Dirichlet-Neumann half-plane diffraction
{title2=$\beta(\alpha)=\sqrt{k^2-\alpha^2}$}
For a scalar wave above a line with a Dirichlet half and a Neumann half, the boundary transform factors into $\beta_+(\alpha)=\sqrt{k+\alpha}$ and $\beta_-(\alpha)=\sqrt{k-\alpha}$. For $e^{-i\omega t}$, the <limiting absorption principle> puts $-k$ below and $k$ above the inversion contour. With Dirichlet conditions on the negative half-line and incident tangential wavenumber $a$, pole subtraction gives the diffracted trace $U_+=2i\sqrt{k-a}/[(\alpha+a)\sqrt{k+\alpha}]$. Its large-transform decay gives a bounded field with a square-root edge trace. The normal derivative has an inverse-square-root singularity on the Dirichlet half; it is identically zero on the Neumann half.
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