Similarly, the interior cylindrical multipole expansion has the functional form Φ Φ ( ρ ρ , θ θ ) = − − Q 2 π π ϵ ϵ ln ⁡ ⁡ ρ ρ ′ + 1 2 π π ϵ ϵ ∑ ∑ k = 1 ∞ ∞ ρ ρ k [ I k cos ⁡ ⁡ k θ θ + J k sin ⁡ ⁡ k θ θ ] where the moments are defined Q = ∫ ∫ d θ θ ′ d ρ ρ ′ ρ ρ ′ λ λ ( ρ ρ ′ , θ θ ′ ) I k = 1 k ∫ ∫ d θ θ ′ d ρ ρ ′ cos ⁡ ⁡ k θ θ ′ ( ρ ρ ′ ) k − − 1 λ λ ( ρ ρ ′ , θ θ ′ ) J k = 1 k ∫ ∫ d θ θ ′ d ρ ρ ′ sin ⁡ ⁡ k θ θ ′ ( ρ ρ ′ ) k − − 1 λ λ ( ρ ρ ′ , θ θ ′ )