If the zeros of a holomorphic function $f$ have an accumulation point, then $f$ is constant.

Question

I'm having trouble with the proof of the next statement:

Let $U\subset \mathbb{C}$ be open and connected. If $f$ is defined in $U$ and is holomorphic, and the set of its zeros have and accumulation point, then $f$ is constant on $U$.

I know how to prove this when that accumulation point is in $U$, but I don't know what to do in the other case. Hope you can help me, please.

Thank you.

The accumulation point has to be in $U$, otherwise the statement is not true. — TheSilverDoe, Jul 30 '21 at 15:41
@TheSilverDoe: Indeed, if the OP's statement were true, then the topic of a huge amount of research in complex analysis would reduce to trivialities. — Dave L. Renfro, Jul 30 '21 at 15:47
@DaveL.Renfro Yes, and probably, all the maths would collapse. — TheSilverDoe, Jul 30 '21 at 15:48
It seems to me that this is often glossed over in the statement of the theorem. For example, the first paragraph of the Wikipedia article on the identity theorem misstates the theorem, so far as I can see. — saulspatz, Jul 30 '21 at 15:48
@saulspatz: It seems to me that this is often glossed over in the statement of the theorem. --- In fact, less than 2 weeks ago I nearly did this myself (in the statement), and actually did so in the application I was making, in the comments to this answer (which I obviously recalled as I was writing my previous comment here). — Dave L. Renfro, Jul 30 '21 at 15:52
@saulspatz Yes, the statement in the preamble to the linked Wikipedia article is sloppy, but if you read the actual statement of the lemma down below, they clarify that the accumulation point lies in the domain. — Moishe Kohan, Jul 30 '21 at 19:00

score 8 · Accepted Answer · answered Jul 30 '21 at 15:39

8

It's not true. For example, $\sin(1/z)$ with $U = \mathbb C \backslash \{0\}$.

answered Jul 30 '21 at 15:39

Robert Israel

470,583

If the zeros of a holomorphic function $f$ have an accumulation point, then $f$ is constant.

1 Answers1