= Solution
For an <axisymmetric harmonic function>, put $\Psi=R(r)\Theta(\theta)$ in the <Laplacian in spherical coordinates> and multiply by $r^2/(R\Theta)$. <Separation of variables> gives
$$
\frac{(r^2R')'}R=-\frac{(\sin\theta\,\Theta')'}{\sin\theta\,\Theta}=\ell(\ell+1).
$$
With $u=\cos\theta$, the angular equation becomes the <Legendre differential equation>. Regularity at both poles selects the <Legendre polynomials> $P_\ell(u)$ with integer $\ell\ge0$; the singular second solutions are excluded. The radial <Euler-Cauchy equation> has solutions $r^\ell$ and $r^{-\ell-1}$. <Superposition> therefore gives
$$
\Psi=\sum_{\ell=0}^\infty(A_\ell r^\ell+B_\ell r^{-\ell-1})P_\ell(\cos\theta),
$$
with coefficients chosen to ensure convergence and the required boundary regularity. The <axisymmetric harmonic extension from a sphere> gives these coefficients directly from boundary data.
Back to article page
