Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

It is a basic fact that the zero set of a non zero holomorphic function defined on a open set $A$ is discrete.

By a result in Rudin's textbook on "Real and Complex Analysis", we know that any real valued harmonic function $u$ is locally the real part of some holomorphic function, so I ask the following question:

Is the zero set $Z(u)$ of a non zero real valued harmonic function $u$ defined on open set $A\subset \mathbb{C}$ discrete?


Note that if $u$ vanishes on a nonempty open set $O\subset A$, then write $f=u+iv$ to be the holomorphic function, then Cauchy-Riemman's theorem implies that $v=0$ on $O$, so $f=0$ on $O$, then $f=0$ on $A$.

share|improve this question
1  
Virtually every harmonic function is a counterexample... –  mrf Jan 8 '13 at 22:52
add comment

1 Answer

up vote 2 down vote accepted

No. The function $f(a+ib)=b$ is harmonic but its zero set is the real line.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.