The general solution to Laplace's equation in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor r, f ( r , θ θ , φ φ ) = ∑ ∑ ℓ ℓ = 0 ∞ ∞ ∑ ∑ m = − − ℓ ℓ ℓ ℓ f ℓ ℓ m r ℓ ℓ Y ℓ ℓ m ( θ θ , φ φ ) , where the f ℓ are constants and the factors r Y ℓ are known as solid harmonics. Such an expansion is valid in the ball r < R = 1 lim sup ℓ ℓ → → ∞ ∞ | f ℓ ℓ m | 1 / ℓ ℓ .