Finite image construction for a planar wedge ($\arg z\in (0,\pi /N),z_k^{+}=z_0{e}^{2\pi ik/N},z_k^{-}=\overline{z_0}{e}^{2\pi ik/N}$)

For a positive integer $N$, a point source in a grounded planar wedge of opening $\pi /N$ can be supplemented by $2N-1$ images. Keep positive sources at $z_0{e}^{2\pi ik/N}$ and negative sources at $\overline{z_0}{e}^{2\pi ik/N}$, $k=0,\dots ,N-1$, counting the actual source as the first positive one. Reflection in either boundary pairs equal distances with opposite signs, so the logarithmic electrostatic potential vanishes on both rays. Every image lies outside the open physical wedge.

The identity ${\prod }_k(z-z_0{e}^{2\pi ik/N})=z^N-z_0^N$ compresses the image sum toHere the normalization is that of a two-dimensional point source, equivalently a charge per unit length in a translationally invariant three-dimensional problem. For $N=3$, a source on the bisector with unit radius has $z_0^3=i$, giving the ratio $(z^3-i)/(z^3+i)$. The method of images works finitely here because successive boundary reflections generate a finite rotation group; arbitrary wedge angles need not have a finite image construction.