 Dirichlet Green function for a three-dimensional quarter-space

For $D=\{x>0,z>0\}\subset {\mathbb{R}}^3$, take the Dirichlet Green function convention $-{\Delta }_{P}G(P,Q)={\delta }_Q(P)$. Reflect $Q=(\xi ,\eta ,\zeta )$ to $Q_{ij}=((-1{)}^i\xi ,\eta ,(-1{)}^j\zeta )$. The displayed image sum vanishes on both faces, has exactly one positive unit source inside $D$, and decays at infinity. Pairwise cancellation proves the boundary conditions, while the three image terms are harmonic inside $D$. Its inward source-normal derivative gives the Poisson kernel of a three-dimensional quarter-space. The source sign must be kept consistent with Green's second identity.