= Dirichlet Green function between parallel planes
{c}
{title2=$G_D(x,y)=-\frac1{4\pi}\sum_{n\in\mathbb Z}\left(|x-y_n^+|^{-1}-|x-y_n^-|^{-1}\right)$}
For a three-dimensional slab $0<x_1<L$ and source $y=(y_1,y_2,y_3)$, define image positions $y_n^+=(y_1+2nL,y_2,y_3)$ and $y_n^-=(-y_1+2nL,y_2,y_3)$. The displayed <method of images> series uses convention $\Delta_xG_D=+\delta(x-y)$. Reflection in $x_1=0$ pairs the plus image $n$ with the minus image $-n$, while reflection in $x_1=L$ pairs it with the minus image $1-n$. Both boundary values therefore vanish. Only $y_0^+=y$ lies in the slab. Each bracket is $O(n^{-2})$, giving absolute convergence of the paired series away from sources. This selects the usual <Dirichlet Green function> that decays at large transverse separation.
Back to article page
