Outgoing Green function for the three-dimensional Helmholtz equation ($G_k(\mathbf{r},{\mathbf{r}}^{\prime })=e^{ik|\mathbf{r}-{\mathbf{r}}^{\prime }|}/(4\pi |\mathbf{r}-{\mathbf{r}}^{\prime }|)$)

The outgoing Green function satisfies $(\Delta +k^2)G_k=-{\delta }_{{\mathbf{r}}^{\prime }}$ and the Sommerfeld radiation condition. For time dependence $e^{-i\omega t}$, its radial phase travels outward. Away from its source it solves the homogeneous Helmholtz equation. At large $r$, $G_k=e^{ikr}{e}^{-ik\hat{\mathbf{r}}\cdot {\mathbf{r}}^{\prime }}/(4\pi r)+O(r^{-2})$.

Let $(\Delta +k^2)u=0$ for $x_3>0$, with real $k>0$, $u=O(r^{-1})$ and $r({\partial }_{r}u-iku)\to 0$ uniformly in direction. With sufficient boundary regularity, Green's second identity on an expanding half-ball gives the title formula. The plane integral is interpreted as the limit over expanding disks when it is not absolutely convergent. Its sign uses the upward derivative ${\partial }_{3}u$; the outward normal derivative is $-{\partial }_{3}u$.

For $G=e^{ik|x-y|}/(4\pi |x-y|)$, $G=O(r^{-1})$ and $({\partial }_r-ik)G=O(r^{-2})$ at large $|y|=r$. The hemispherical integrand can be written $u({\partial }_r-ik)G-G({\partial }_r-ik)u$, whose integral tends to zero by the Sommerfeld radiation condition. Add the boundary identities for a source point $x$ and its reflected point $x^{\ast }=(x_1,x_2,-x_3)$. On the plane $G(x,y)=G(x^{\ast },y)$ and their normal derivatives are opposite. The terms involving boundary values of $u$ cancel, leaving twice $G{\partial }_{3}u$ and hence the title formula. In particular, zero Neumann boundary data imply $u=0$ in this radiating class.

For $k_0>0$, $l=|r-r^{\prime }|>0$ and the convention $(\Delta +k_0^2)G=-\delta$, the spatial Fourier transform gives the multiplier $(k^2-k_0^2{)}^{-1}$. Aligning the Fourier polar axis with $r-r^{\prime }$ gives ${\int }_{S^2}{e}^{ikl\cos \theta }d\Omega =4\pi \sin (kl)/(kl)$ and hence the displayed radial integral.

The pole on the integration path requires a prescription. The Cauchy principal value gives $G_{\mathrm{PV}}(l)=\cos (k_{0}l)/(4\pi l)$. The outgoing prescription is $k^2-k_0^2-i0$, yielding $G_{+}(l)=e^{i{k}_{0}l}/(4\pi l)$; the opposite prescription gives the incoming exponential. The Sokhotski–Plemelj formula $1/(q-i0)=\mathrm{PV}(1/q)+i\pi \delta (q)$ shows that the additional radial contribution is $i\sin (k_{0}l)/(4\pi l)$.

For the inverse-kernel convention $(\Delta +k_0^2)H=+\delta$, use $H=-G$. Thus a source equation $(\Delta +k_0^2)\Phi =V$ has $\Phi =H\ast V=-G\ast V$, with the same boundary or radiation convention throughout. Without a pole prescription or an equivalent condition at infinity, the real-frequency source problem does not specify a unique Green function.

For the upper half-space and the sign convention $(\Delta +k^2)G_D={\delta }_{r^{\prime }}$, reflect the source $r^{\prime }=(x^{\prime },y^{\prime },z^{\prime })$ to $r^{\prime \ast }=(x^{\prime },y^{\prime },-z^{\prime })$ and use the displayed difference. This is the negative of the conventional outgoing kernel, whose source sign is $-\delta$. The method of images makes the Dirichlet boundary condition zero on the plane. With $R=|r^{\prime }-r|$ and $\cos \theta =z^{\prime }/R$, its outward normal derivative there isThe outward unit normal is $-e_z$. Differentiating $e^{ikR}/R$ gives $e^{ikR}(ik/R-1/R^2)$; the real and reflected source contributions add in the normal derivative. Green's second identity therefore expresses a solution of the Helmholtz equation as $\varphi (r^{\prime })={\int }_{z=0}\varphi (r){\partial }_n{G}_D(r,r^{\prime })dS$ when the boundary term at infinity vanishes. Suitable decay or compatible outgoing Sommerfeld radiation conditions are needed; vanishing at infinity without further control is not a general uniqueness condition for a real-frequency Helmholtz problem.

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.