The spherical multipole expansion takes a simple form if the charge distribution is axially symmetric (i.e., is independent of the azimuthal angle ϕ ϕ ′ ). By carrying out the ϕ ϕ ′ integrations that define Q ℓ ℓ m and I ℓ ℓ m , it can be shown the multipole moments are all zero except when m = 0 . Using the mathematical identity P ℓ ℓ ( cos ⁡ ⁡ θ θ ) = d e f 4 π π 2 ℓ ℓ + 1 Y ℓ ℓ 0 ( θ θ , ϕ ϕ ) the exterior multipole expansion becomes Φ Φ ( r ) = 1 4 π π ε ε ∑ ∑ ℓ ℓ = 0 ∞ ∞ ( Q ℓ ℓ r ℓ ℓ + 1 ) P ℓ ℓ ( cos ⁡ ⁡ θ θ ) where the axially symmetric multipole moments are defined Q ℓ ℓ = d e f ∫ ∫ d r ′ ρ ρ ( r ′ ) ( r ′ ) ℓ ℓ P ℓ ℓ ( cos ⁡ ⁡ θ θ ′ ) In the limit that the charge is confined to the z -axis, we recover the exterior axial multipole moments.