Is it possible to explicitly find a harmonic function $u \in C^2(\mathbb{R}^2)$ such that \begin{equation}\tag{$\dagger$}\label{eq:dag} u(x,1) = u(x,-1) = 0 \end{equation} for all $x \in \mathbb{R}$? Is it possible for $u$ to be a polynomial?
My intuition says that $u$ cannot be a polynomial, but I have been unable to rigorously prove this. In fact, I haven't been able to come up with a function satisfying \eqref{eq:dag}.
One such function is $$ u(x,y)=\Re\bigl(-i\,e^{\pi(x+iy)}\bigr)=e^{\pi\,x}\sin(\pi\,y). $$ If $u$ is such a function, then $u=\Re(f(z))$ for some entire function $f$, with $f(x+i)=f(x-i)=0$. Schwarz's reflection principle implies that $f$ must be periodic of period $2\,i$, and hence it cannot be a polynomial.