Bôcher's theorem

Remarks

So an isolated singularity of a positive harmonic function is a nonnegative multiple of the fundamental solution of the Laplacian, up to a harmonic term. If $b = 0$, the singularity is removable, in agreement with the removable singularity theorem. Proof idea: averaging $u$ over the spheres about $0$ gives a radial harmonic function on the punctured ball. It has the form $a + b\abs{x}^{2-n}$, or $a + b \log(1/\abs{x})$ in the plane. Positivity of $u$ is then used to show that $b \ge 0$ and that the difference of $u$ and the singular term has a removable singularity at $0$. Positivity is needed: $u(x) = x_1 / \abs{x}^n$ is harmonic on $\R^n \setminus \set{0}$, takes both signs, and is not of the stated form, since it grows like $\abs{x}^{1-n}$ near $0$. Example: $u(x) = 1 + \abs{x}^{2-n}$ is positive on the punctured unit ball for $n \ge 3$, with $v = 1$ and $b = 1$.

Going deeper

For a student

The theorem classifies isolated singularities of positive harmonic functions, and it is the local form of the Riesz decomposition of positive superharmonic functions. A global corollary: a positive harmonic function on $\R^n \setminus \set{0}$, $n \ge 3$, has the form $a + b\abs{x}^{2-n}$ with $a, b \ge 0$, by combining Bôcher's theorem with the Kelvin transform and Liouville's theorem. It is also the model for the description of the Martin boundary at an isolated boundary point.