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 $r$, $G_k=e^{ikr}{e}^{-ik\hat{\mathbf{r}}\cdot {\mathbf{r}}^{\prime }}/(4\pi r)+O(r^{-2})$.