The generalization to an arbitrary distribution of line charges λ λ ( ρ ρ ′ , θ θ ′ ) is straightforward. The functional form is the same Φ Φ ( r ) = − − Q 2 π π ϵ ϵ ln ⁡ ⁡ ρ ρ + 1 2 π π ϵ ϵ ∑ ∑ k = 1 ∞ ∞ C k cos ⁡ ⁡ k θ θ + S k sin ⁡ ⁡ k θ θ ρ ρ k and the moments can be written Q = ∫ ∫ d θ θ ′ d ρ ρ ′ ρ ρ ′ λ λ ( ρ ρ ′ , θ θ ′ ) C k = 1 k ∫ ∫ d θ θ ′ d ρ ρ ′ ( ρ ρ ′ ) k + 1 λ λ ( ρ ρ ′ , θ θ ′ ) cos ⁡ ⁡ k θ θ ′ S k = 1 k ∫ ∫ d θ θ ′ d ρ ρ ′ ( ρ ρ ′ ) k + 1 λ λ ( ρ ρ ′ , θ θ ′ ) sin ⁡ ⁡ k θ θ ′ Note that the λ λ ( ρ ρ ′ , θ θ ′ ) represents the line charge per unit area in the ( ρ ρ − − θ θ ) plane.