Finding all harmonic radial functions

Question

Find all harmonic functions in $\mathbb{C}\setminus\{0\}$ which are constant on the circles $$ \{ z \in\mathbb{C} : |z| = r \}. $$ How to start finding these functions?

Answer 1

$f(r,\theta)$ constant on circles $\implies \frac{\partial f}{\partial\theta} = 0$. Express the Laplacian in terms of $r$ and $\theta$, and you find Laplace's equation reduces to solving $$ 0 = \frac{1}{r}\frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right). $$

Answer 2

The only radial harmonic functions on $\mathbb {C}\setminus \{0\}$ are of the form $a + b\log |z|,$ where $a,b$ are constants. Proof: Suppose $u$ is radial and harmonic on $\mathbb {C}\setminus \{0\}.$ Then

$$v(z) = u(1) +\frac{u(2)-u(1)}{\log 2}\cdot \log |z|$$

is radial and harmonic on $\mathbb {C}\setminus \{0\}.$ We have $u=v$ on the boundary of $\{1<|z|<2\},$ so by the maximum principle, $u=v$ on $\{1<|z|<2\}.$ This implies $u=v$ on $\mathbb {C}\setminus \{0\}$ by the local maximum principle.