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

  • $\begingroup$ A function "vanishes" on a set means if the function takes the value $0$ (or additive identity, more generally) on the set. $\endgroup$ – Theo Bendit Jul 30 at 2:00
  • $\begingroup$ I edited the question. So does that mean $Re(u) = 0?$ $\endgroup$ – Overachiever Jul 30 at 2:05
  • $\begingroup$ No, it means $u(x, 0) = 0$ for all $x$. $\endgroup$ – Theo Bendit Jul 30 at 2:06
  • $\begingroup$ @TheoBendit, where $u(x, y) = u(x + iy)$. Just to re-phrase, $u(z)=0$ when $Im(z)=0$, i.e. $z \in \mathbb{R}$. $\endgroup$ – Eric Canton Jul 30 at 4:06
  • $\begingroup$ The zero function and your function $iy$ both vanish on the real line. That answers your question. $\endgroup$ – Kabo Murphy Jul 30 at 5:25

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