The question: Give two different harmonic functions on $\mathbb{C}$ that vanish on the entire real axis.
Now I don't understand what they mean by vanish. Does that mean $Re(u) = 0$
The simplest functions I can think of is a coordinate harmonic function so like $u(x,y) = iy$. However, I think I am not thinking this right. How do I approach this??
Please do not solve the problem completely. Only give me hints.
Thank you very much