for some number m. A priori, m is a complex constant, but because Φ must be a periodic function whose period evenly divides 2 π, m is necessarily an integer and Φ is a linear combination of the complex exponentials e. The solution function Y (θ, φ) is regular at the poles of the sphere, where θ = 0, π. Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ (ℓ + 1) for some non-negative integer with ℓ ≥ | m | ; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial P ℓ (cos θ) . Finally, the equation for R has solutions of the form R (r) = A r + B r ; requiring the solution to be regular throughout R forces B = 0.