Here the solution was assumed to have the special form Y (θ, φ) = Θ(θ) Φ(φ). For a given value of ℓ, there are 2 ℓ + 1 independent solutions of this form, one for each integer m with − ℓ ≤ m ≤ ℓ. These angular solutions are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Y ℓ ℓ m ( θ θ , φ φ ) = N e i m φ φ P ℓ ℓ m ( cos ⁡ ⁡ θ θ ) which fulfill r 2 ∇ ∇ 2 Y ℓ ℓ m ( θ θ , φ φ ) = − − ℓ ℓ ( ℓ ℓ + 1 ) Y ℓ ℓ m ( θ θ , φ φ ) .