Dirichlet Green function inside a sphere ($G_D(x,y)=-\frac{1}{4\pi }\left[|x-y{|}^{-1}-\frac{a}{|y|}|x-a^{2}y/|y{|}^2{|}^{-1}\right]$)

For the three-dimensional ball $|x|<a$, the displayed Dirichlet Green function has convention ${\Delta }_x{G}_D=+\delta (x-y)$. Its method of images uses the exterior inverse point $y^{\ast }=a^{2}y/|y{|}^2$. On $|x|=a$, the distance identity $|x-y^{\ast }|=(a/|y|)|x-y|$ makes the boundary value zero. The image lies outside the ball and therefore adds no source singularity to the domain. At a central source the weighted image term tends to $1/a$, giving $G_D(x,0)=-(4\pi |x|{)}^{-1}+(4\pi a{)}^{-1}$. Spherical inversion with the required radial weight is the Kelvin transform.

For the upper half of a three-dimensional ball, let $\overline{y}$ be the reflection of an interior source $y$ in the equatorial plane. The Dirichlet Green function inside a sphere, evaluated at the source and its reflection and then subtracted, vanishes on both the spherical cap and flat equatorial disc. Reflection makes the source distances and image distances agree on the plane. Only the original source is in the upper half-ball; the reflected source is in the lower half-ball, and both inversion images are outside the ball. Thus the difference has precisely the required Dirac delta distribution and Dirichlet boundary conditions.