Herglotz wave function
= Herglotz wave function
{c}
A Herglotz wave function is a superposition of incident <plane waves>,
$$
v_g(\mathbf r)=\int_{S^2}e^{ik\widehat{\mathbf d}\cdot\mathbf r}g(\widehat{\mathbf d})\,dS(\widehat{\mathbf d}),\qquad g\in L^2(S^2).
$$
It solves the <Helmholtz equation> everywhere and is <real analytic>. The finite sphere area makes the density integrable, so each spatial <derivative> can be taken under the integral.