which can be written as Φ Φ ( r ) = 1 4 π π ε ε ∑ ∑ ℓ ℓ = 0 ∞ ∞ ∑ ∑ m = − − ℓ ℓ ℓ ℓ ( Q ℓ ℓ m r ℓ ℓ + 1 ) 4 π π 2 ℓ ℓ + 1 Y ℓ ℓ m ( θ θ , ϕ ϕ ) where the multipole moments are defined Q ℓ ℓ m = d e f q ( r ′ ) ℓ ℓ 4 π π 2 ℓ ℓ + 1 Y ℓ ℓ m ∗ ∗ ( θ θ ′ , ϕ ϕ ′ ) .