Similarly, the interior multipole expansion has the same functional form. In the interior case, where r ′ > r , the result is: Φ Φ ( r ) = 1 4 π π ε ε ∑ ∑ ℓ ℓ = 0 ∞ ∞ ∑ ∑ m = − − ℓ ℓ ℓ ℓ I ℓ ℓ m r ℓ ℓ 4 π π 2 ℓ ℓ + 1 Y ℓ ℓ m ( θ θ , ϕ ϕ ) , with the interior multipole moments defined as I ℓ ℓ m = d e f ∫ ∫ d r ′ ρ ρ ( r ′ ) ( r ′ ) ℓ ℓ + 1 4 π π 2 ℓ ℓ + 1 Y ℓ ℓ m ∗ ∗ ( θ θ ′ , ϕ ϕ ′ ) .