For clarity, we illustrate the expansion for a single line charge, then generalize to an arbitrary distribution of line charges. Through this article, the primed coordinates such as ( ρ ρ ′ ′ , θ θ ′ ′ ) refer to the position of the line charge(s), whereas the unprimed coordinates such as ( ρ ρ , θ θ ) refer to the point at which the potential is being observed. We use cylindrical coordinates throughout, e.g., an arbitrary vector r has coordinates ( ρ ρ , θ θ , z ) where ρ ρ is the radius from the z axis, θ θ is the azimuthal angle and z is the normal Cartesian coordinate. By assumption, the line charges are infinitely long and aligned with the z axis.