 Solution

Reflect the physical source across $z=0$, writing $\overline{{\mathbf{r}}^{\prime }}=(x^{\prime },y^{\prime },-z^{\prime })$. Its radius is still $s=|{\mathbf{r}}^{\prime }|$, and its spherical image is $\overline{{\mathbf{r}}^{\ast }}$. The Dirichlet Green function inside a hemisphere isExplicitly this four-source method of images isBoth $G_0$ terms vanish on the spherical cap. On the flat equatorial disc, reflection equates the two physical-source distances and also the two image-source distances, so both pairs cancel. The reflected physical source lies in the lower half-ball and the two inverted images lie outside the sphere. Consequently the only source in the upper half-ball is ${\mathbf{r}}^{\prime }$, with the correct coefficient:The Dirichlet uniqueness for the Poisson equation then identifies this as the required Green function.