Harmonic functions which vanish on the real line

Question

Let $u:\mathbb{C}\to \mathbb{R}$ be harmonic on the entire plane. Is there a nice classification, maybe a nice basis, for the set of such functions that satisfy $u\equiv 0$ on $\mathbb{R}$?

Examples:

• $u(x,y)=0$
• $u(x,y)=y$
• $u(x,y)=\mathrm{Im}( (x+iy)^2)=2xy$

Answer 1

Say $v$ is a harmonic conjugate of $u$. Then $f=iu-v$ is an entire function, so $f(z)=\sum a_n z^n$. Now saying $u=0$ on $\Bbb R$ says that $f$ is real-valued on $\Bbb R$, which is easily seen to imply that $a_n$ is real. So:

$u$ is a real-valued harmonic function in $\Bbb C$ vanishing on $\Bbb R$ if and only if there exists a sequence $(a_n)$ of real numbers with $|a_n|^{1/n}\to0$ and $u(z)=\Im\sum a_nz^n.$