= Dirichlet Green function inside a sphere
{c}
{title2=$G_D(x,y)=-\frac1{4\pi}\left[|x-y|^{-1}-\frac a{|y|}|x-a^2y/|y|^2|^{-1}\right]$}
For the three-dimensional ball $|x|<a$, the displayed <Dirichlet Green function> has convention $\Delta_xG_D=+\delta(x-y)$. Its <method of images> uses the exterior inverse point $y^*=a^2y/|y|^2$. On $|x|=a$, the distance identity $|x-y^*|=(a/|y|)|x-y|$ 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>.
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