Weyl plane-wave representation
= Weyl plane-wave representation
{c}
{title2=$\frac{e^{ikR}}R=\frac{ik}{2\pi}\int\frac{e^{ik[p(x-x')+q(y-y')+m|z-z'|]}}m\,dp\,dq$}
Here $m=\sqrt{1-p^2-q^2}$ has nonnegative imaginary part. This representation of the outgoing <Helmholtz equation> kernel separates propagating <plane waves> from <evanescent waves>. The absolute vertical separation is required for fields on either side of a source; choosing its sign fixes the upward or downward angular spectrum.