Let $h$ be a harmonic polynomial that is zero on the lines $Im(z) = 1$ and $Im(z) = -1$.
I know that by the Schwarz Reflection Principle, $h$ must be zero on any line $Im(z) = k$ for $k$ odd, but this fact might not be necessary.
I'm pretty sure that $h$ must be identically zero, but how do I prove this statement?
I've been assuming that a harmonic polynomial is simply an element of $\Bbb R[x,y]$ whose Laplacian is zero, but I've come across expressions like $f(z) = z^n + \overline z^n$, and I'm not sure how those fit into the picture.
I'm just starting to get into harmonic function theory, so I apologize if my questions are very basic. That said, I'll be very grateful for any pointers that you can provide. Thanks!